+
    i:                        ^ RI Ht ^RIHt ^RIHtHtHt ]RR l4       t]RR l4       t	]R 4       t
]R 4       t. RNR	NR
NRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNR NR!NR"NR#NR$NR%NR&NR'NR(NR)NR*NR+NR,NR-NR.NR/NR0NR1NR2NR3NR4NR5NR6NR7NR8NR9NR:NR;NR<NR=NR>NR?NR@NRANRBNRCNRDNRENRFNRGNRHNRINRJNRKNRLNRMNRNNRONRPNRQNRRNRSNRTNRUNRVNRWNRXNRYNRZNR[NR\NR]NR^NR_NR`NRaNRbNRcNRdNReNRfNRgNRhNRiNRjNRkNtRl t]RRm l4       t]Rn 4       t]Ro 4       t]Rp 4       t]Rq 4       t]Rr 4       t]Rs 4       t]RRu l4       tRv tRw tRx tRy t]Rz 4       t]RR{ l4       t]RR| l4       t]R} 4       t]R~ 4       t]RR l4       t]RR l4       t R t!R t"]^ .Rt3R l4       t#]^.^ 3R l4       t$R t%R t&R t'R t(]RR l4       t)]R 4       t*R# )    )print_function)xrange)defundefun_wrappeddefun_staticc                   a aaa	 S P                  S4      oS P                  S4      oS^ 8  d   S P                  R4      # \        S R4      '       d   S P                  pM	/ ;pS n        S^8X  d;   S^ 8X  d   S P                  5# SV9   d    VS,          w  rEVS P
                  8  d   V5# ^o	VV V	V3R lpS P
                  p S^28  d(   ^S n        S P                  V^ S P                  .^R7      o	V^
,           \        SR,          4      ,           S n        S P                  V^ S P                  .^R7      pS P                  S4      S,          ^S,          ,          S P                  S4      S^,           ,          S^,           ,          ,
          ^V,          S,          S	,          ,           pVS n        S^8X  d)   S P                  S4      '       d   S P
                  V3VS&   V5#   TS n        i ; i)r   z&Stieltjes constants defined for n >= 0stieltjes_cachec                 j  < V S,          pVSP                   ,
          SP                  SSP                   V ,          ,
          4      S,          ,          ^V^,          ,           ,          SP                  ^SP                  ,          V ,          4      ^,
          ,          pSP	                  V4      S,          #    )jlnexppi_re)xxavactxmagns   &  u/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/mpmath/functions/zeta.pyfstieltjes.<locals>.f   sw    qSXsvvaai(!++Qr1uW5swwqxz7J17LMwwqzC    )	maxdegree      ?)convert
bad_domainhasattrr	   eulerprecquadinfintr   isint)
r   r   r   r	   r#   sr   origr   r   s
   fff      @r   	stieltjesr*      s   AAAA1u~~FGGs%&&--022#-Av6II:%a(GDsxxr	
C    88D
 r6CH((1qkQ(7C"9s1c6{*HHQ377rH2FF1IqL!A#QqS!11Q3!77!A#a%)CAv#))A,,!hh]2I s    C'G   	G)c                    \        V4      pWP                  8X  g   WP                  8X  dC   V^8  d0   WP                  8X  d   V^ 8X  d   V P                  # V P                  # V P                  # V^ 8X  Ed	   V P	                  V4      '       d}   V P                  RRV,          ,           4      pV P                  RRV,          ,
          4      pV P                  V P                  4      ) ^,          V,          RWE,
          ,          ,
          # V P                  V4      '       d   V# V P	                  V P                  RRV,          ,           4      4      V P                  V P                  4      ^,          V,          ,
          # V^ 8  EdF   RV^,
          ,          V P                  V^,
          RRV,          ,
          4      ,          pRV^,
          ,          V P                  V^,
          RRV,          ,           4      ,          pV P	                  V4      '       dM   V^8X  d7   RV P                  V P                  4      ,          RWE,           ,          ,           # RWE,           ,          # V^8X  dF   V P                  RV P                  V P                  4      ,          RWE,           ,          ,           4      # V P                  RWE,           ,          4      # R# )         ?y              ?N                   )r&   r%   ninfzero_imloggammar   r   isinf	polygammalogr   )r   t
derivativedr   bs   &&&   r   siegelthetar;   ,   s   JA	
gghhq5HH}axx77N88OAv771::T$q&[)AT$q&[)AFF366N?1$Q&qs33yy||773<<T!V45svvq8H8JJJ1uacN3==1d46k::QqSM#--!T$q&[99771::AvCGGCFFO+D!#J66QSz!AvwwtCGGCFFO3D!#J>??wwtQSz** r   c           	         a a ^S P                   ,          S P                  ^S P                  ^S,          ^,           ^S P                  ,          ,          4      ,           4      ,          pS P	                  V V3R lV4      # )r,   c                 V   < SP                  V 4      SP                  S,          ,
          # N)r;   r   )r7   r   r   s   &r   <lambda>grampoint.<locals>.<lambda>S   s    #//!"4SVVAX"=r   )r   r   lambertwefindroot)r   r   gs   ff r   	grampointrE   N   sU     	
#&&3<<1Q355(9::;;A<<=qAAr   c           
      	  a aaaaaaaaaaaa \        VP                  R ^ 4      4      pS P                  V4      pS P                  V4      pS P	                  V4      pS P
                  p \        V4      RV,          8  dJ   V^,          V8  d<   S P                  W4      pS P                  V4      '       d   S P                  V4      # V# S ;P
                  ^,          un        S P                  S P                  T4      4      pS P                  RS P                  T,          ,           4      oT^ 8X  d<   TS,          pTS n        S P                  T4      '       d   S P                  T4      # T5# S P                  RS P                  T,          ,           ^R7      oS P                  T^R7      oT^8X  d[   S P                  T,          SSS,          ,           ,          pTS n        S P                  T4      '       d   S P                  T4      # T5# S P                  RS P                  T,          ,           ^R7      oS P                  T^R7      oS^,          S P                  S,          ,
          oT^8X  dY   TTTTT3R lp	S P                  T	^4      pT) T,          pTS n        S P                  T4      '       d   S P                  T4      # T5# S ;P
                  ^
,          un        S P                  RS P                  T,          ,           ^R7      oS P                  T^R7      oS^,          ^S P                  ,          S,          S,          ,
          S,
          oT^8X  dl   TTTTTTT3R lp	S P                  T	^4      pS P                  ) T,          T,          pTS n        S P                  T4      '       d   S P                  T4      # T5# S P                  RS P                  T,          ,           ^R7      oS P                  T^R7      oT TTTT3R lp	S P                  T	^4      oT^8X  d\   TTT TTTTTTT3
R lp	S P                  T	^4      pY,          pTS n        S P                  T4      '       d   S P                  T4      # T5# T^8  d!   T 3R lp
S P                  YT^,
          R	7      # R
#   \         d     ELi ; i)r8     r   r8   c                  8   < ^S,          S,          SSS ,          .# r,    )comb1theta1zz1z2s   r   termssiegelz.<locals>.termsz   s    bDKQuW--r   c                  b   < ^S,          S,          ^S,          S ,          SSS,          ,           .# )   rK   )rL   comb2rM   rN   rO   rP   z3s   r   rQ   rR      s(    vXb[!B$u*b5j99r   c                     < S^,          RS P                   ,          S^,          ,          S,          RS^,          ,          RS,          S,          S P                   S,          .# )   r   )r   rM   theta2theta3theta4s   r   rQ   rR      sL    	2cee8FAI-f4blvIfceeFl, 	,r   c                     <
 ^S^,          ,          S,          RSP                   ,          S,          S,          ^S,          S,          ^S,          S ,          S	SS,          .# )   rY   r\   )
rU   comb3r   rM   r]   rN   rO   rP   rV   z4s
   r   rQ   rR      sL    vqy[^RXb[%76"2eR5* *r   c                 *   < SP                  V ^R7      # )rX   rH   )siegelz)r   r   s   &r   r?   siegelz.<locals>.<lambda>   s    ckk!k2r   )r   N)r&   getr   r   r2   r#   absrs_z_is_real_typeNotImplementedErrorexpjr;   zetar   sum_accuratelydiff)r   r7   kwargsr9   t1t2r#   r   e1rQ   hrL   rU   rb   rM   r]   r^   r_   rN   rO   rP   rV   rc   s   f&,        @@@@@@@@@@@@r   re   re   V   s   FJJ|Q'(AAA	B	B88Dr7SX"a%"*A  ##wwqz!H HHNH	#//!$	%BSUU1WAAvqDQ771:r		#ceeAg+!	,B__Q1_-FAvUU2Xr!F({#Q771:r		#ceeAg+!	,B__Q1_-FAIceeFl"EAv	. 	.ua(SUQ771:r	HHNH	#ceeAg+!	,B__Q1_-FAIagfnV++F2EAv	: 	:ua(eeVBYq[Q771:r		#ceeAg+!	,B__Q1_-F, , ua(EAv	* 	* ua(TQ771:r	1u2xx!x$$ y  s   )R.  AR. R. .R=<R=gD,@g|c5@g9@gll>@gYw@@gjDB@gp`uD@g,ݩE@geȨ H@gPH@gN~3|J@gC 9L@g7kM@g1wjN@g?a3GP@gP@gM>bQ@gY`OLR@gR@gXGEIS@gXeS@giCT@gH&Q/U@gܩ7U@gK`z3V@gwG{W@gꞯW@grxW@g˻;I2X@gSWTY@gts7oY@g~x\Z@gEAZ@gK[@g|j[@gZ~\@gU-]@gB/]@gg W^@g'٘^@go_@gCY_@gڡ2`@g#-Xb`@ggv`@gS5`@gӝCa@g{wa@g:ia@g=a@gl@b@g'B/Kmb@g;upb@gHٵb@gլJ c@gc@gZUxc@gҍd3c@gu&d@g`d@gĸ/d@g2d@g1E#e@g L.=e@gOТN+e@gBV"e@g': f@gxLf@gKS}f@gcf@gg@goLc<)3g@gstUSgg@g.+Qg@g6p^ h@g"h@g%-!~hh@g@T#h@goj}h@g
x(i@g㸆Oi@gxۙi@g5\i@gk i@g늁r2j@gqߋvj@g'"j@g֍dj@ghlk@gѿ)bk@gyk@gWȭk@g=IX9 l@g `Lewl@gx|ml@g+P̪l@gl@gbnl@g]06m@gJ}Ɛm@c                     ^ RI pVP                  V 4      pVP                  4        Uu. uF  p\        V4      NK  	  pp\	        V^ ,          4      ^8X  g   Q hV\
        R&   R# u upi )r   N:NNN)urlliburlopen	readlinesfloatround_zeta_zeros)urlrv   r9   r   Ls   &    r   _load_zeta_zerosr~      sU    sA;;=)=aq=A)1;"KN 	*s   A"c           	         \        V4      pV^ 8  d!   V P                  V) 4      P                  4       # V^ 8X  d   \        R4      hV\	        \
        4      8  d   VR8:  d   \        V4       V\	        \
        4      8  d   \        R4      hV P                  RV P                  V P                  \
        V^,
          ,          4      4      # )r   zn must be nonzeroi zn too large for zetazerosr   )r&   zetazero	conjugate
ValueErrorlenr{   r~   rk   mpcrC   re   )r   r   r|   s   &&&r   oldzetazeror      s    AA1u||QB))++Av,--3{V3{!"=>>773S[[+ac2BCDDr   c           	         V^ 8X  d   V P                   # \        V4      R8  de   V P                  V4      pRV P                  V P                  V4      4      ,          p\        V4      \        V4      V P                  ,          8  d   V# \        V4      R8  d=   V ;P
                  \        V P                  \        V4      ^4      ) 4      ,          un        V P                  ;rEV P                  V4      p^p\        V4      \        V4      V P                  ,          8  dF   WV,          V,          pWEWpP                  V^,           4      ,          ,          ,          pV^,          pKo  V# )r     r   g{Gz?)r1   rh   lisqrtepsr#   r&   r6   oner   	_zeta_int)r   r   r   r:   r(   r7   uks   &&      r   riemannrr      s    Avxx
1v}FF1Isxx{##q6CF377N"H
1v}CQ**++GGOAq	A	A
a&3q6#''>
!EAI	!mmAaC(())	QHr   c                 ^    \        V4      pV^8  d   ^ # \        V P                  V4      4      # rJ   )r&   r   list_primes)r   r   s   &&r   primepir      s)    AA1usq!""r   c                    \        V4      pV^8  d   V P                  P                  # VR8  d+   V P                  P                  V P	                  V4      4      # V P                  V4      pV P                  VRR7      V P                  VRR7      ,          ^,          V P                  RR7      ,          pV P                  V P                  P                  V4      V,
          P                  RR7      pV P                  V P                  P                  V4      V,           P                  RR7      pV P                  P                  WE.4      # )r,   ia
  r   )roundingr9   )r&   _ivr1   mpfr   r   r   r   r   floorr   ceilr:   )r   r   miderrr   r:   s   &&    r   primepi2r      s    AA1uww||4xww{{3;;q>**
&&)C
((1c(
"366!S6#9
9!
;CFFCF<P
PC		377;;s#C'**S	9A#''++c"3&))C8A77;;ur   c                   a aa S P                  S4      '       d   S# S P                  S4      ^ 8:  d   \        R4      hS^8X  d   S P                  # SR8X  d'   S P	                  S P
                  S P                  4      # S P                  S4      pVS P                  8  d
   RS,          # S P                  \        V4      ,           oV VV3R lpS P                  V4      # )r   z.prime zeta function defined only for re(s) > 0r   c               3     <"   SP                   p ^ p V^,          pSP                  V4      pV'       g   K%  SSn         VSP                  SP                  VS,          4      4      ,          V,          pV'       g   R# V Sn         Vx  Kx  5i)r   N)r#   moebiusr   rm   )r)   r   r   r7   r   r(   wps       r   rQ   primezeta.<locals>.terms  sr     88D AQKKNcffSXXac]++A-s   1B
AB
;B
)
isnanrer   r%   r   r0   r   r#   r&   rn   )r   r(   rrQ   r   s   ff  @r   	primezetar      s    
yy||
vvayA~IJJAvwwCxwwsxx((q	A388|AvXXA	" e$$r   c                 
  a aa \        S4      oS^ 8  d   \        R4      hS^ 8X  g   S^8X  d   S^8  d   S P                  S4      # SR8X  d7   S P                  ^^S,
          4      ^,
          S P                  S4      ,          # S^8:  dr   S^ 8X  d
   S^ ,          # S^8X  d
   SR,
          # S^8X  d&   ^S,          S^,
          ,          ^,           ^,          # S^8X  d   SSSR,
          ,          R,           ,          # S P	                  S4      '       d
   SS,          # S P                  S4      '       d   S# \        S4      ^8  d(   V VV3R lpS P                  V4      SS,          ,          # V VV3R lpS P                  V4      # )r   z-Bernoulli polynomials only defined for n >= 0r   g      ?c               3   (  <"   SP                   p V x  SP                   S,          p^pVS8:  da   V S^,           V,
          ,          V,          V,          p V^8  d   V^,          '       g   V SP                  V4      ,          x  V^,          pKg  R# 5ir   N)r   	bernoulli)r7   r   r   r   r   rN   s      r   rQ   bernpoly.<locals>.terms6  sw     AG	AAq&qs1uIaKMA!a%%CMM!,,,Q	 s   A'B+'Bc               3   2  <"   SP                  S4      x  SP                  p ^pVS8:  dj   V S^,           V,
          ,          V,          S,          p SV,
          pV^8  d   V^,          '       g   V SP                  V4      ,          x  V^,          pKp  R# 5ir   )r   r   )r7   r   mr   r   rN   s      r   rQ   r   B  s}     --""AAq&qs1uIaK!OaCA!a%%CMM!,,,Q s   A,B0'B)r&   r   r   ldexpr4   r   rh   rn   )r   r   rN   rQ   s   fff r   bernpolyr   !  s@    	AA1uHIIAv!q&QU}}QCx		!AaC "CMM!$444Av6!q&=6!c'>61Q3!9Q;/)6!Q#Ys]++
yy||Av
yy||
1vz		 !!%(1a4//		 !!%((r   c                   a aa \        S4      oS^ 8  d   \        R4      hS^8:  d8   S^ 8X  d
   S^ ,          # S^8X  d
   SR,
          # S^8X  d   SS^,
          ,          # S P                  S4      '       d
   SS,          # S P                  S4      '       d   S# S^,           pS^ 8X  dL   RS P	                  ^V4      ^,
          ,          S P                  V4      ,          V,          S^ ,          ,          # S^8X  dL   ^S P	                  ^V4      ^,
          ,          S P                  V4      ,          V,          S^ ,          ,          # SR8X  dz   S^,          '       d   S P                  # S^d8  g6   SS P                  RS,          4      ,          S P                  R,          8  d#   S P	                  S P                  S4      S) 4      # V VV3R lpS P                  V4      V,          # )r   z)Euler polynomials only defined for n >= 0r   g.eT>?r-   c               3   |  <"   SP                   p ^ pSP                  ^S^,           4      p SV,
          ^,           pV^8  d   V^,          '       g)   ^V,
          SP                  V4      ,          V ,          x  V^,          pVS8  d   R# V S,          SV,
          ^,           ,          V,          p VR,          pK  5i)r   r   N)r   r   r   )r7   r   wr   r   r   rN   s       r   rQ   eulerpoly.<locals>.termsg  s     GGIIa!!AAEa!eesCMM!,,Q..FA1u!QqSUAAHAs   AB<A*B<)r&   r   r4   r   r   r   r1   r   r#   	_eulernumrn   )r   r   rN   r   rQ   s   fff  r   	eulerpolyr   N  s|   AA1uDEEAv6!q&=6!c'>6!QqS'>
yy||!t
yy||	!AAv399Qq>!#$S]]1%55a7!Q$>>Av#))Aa."#CMM!$44Q6A==Cxq5588Os7a
1--=99S]]1-r22 e$q((r   Fc                 $   \        V4      pV'       d   \        V P                  V4      4      # V^d8  d!   V P                  V P                  V4      4      # V^,          '       d   V P                  # V P	                  V P                  VR4      V4      # )d   r   )r&   r   r   r1   r   r   )r   r   exacts   &&&r   eulernumr   v  so    AA3==#$$3wwws}}Q'((1uuxx99S]]1S)1--r   c                     V P                   5pV P                  p^pTp WeV,          ,          pWG,          p\        V4      V8  d    V# Wb,          pV^,          pK>  r   )r   r1   rh   )r   r(   rN   tollr   zkterms   &&&     r   polylog_seriesr     s\    77(CA	A	
B
qDy		t9s? H 		Qr   c                 V   V^ 8  d
   V^ ,          # RV P                   ,          pW1,          ) V P                  V4      ,          V P                  WP                  V4      V,          4      ,          pV P	                  V4      '       d   V^ 8  d   V P                  V4      pV P                  V4      ^ 8  g-   V P                  V4      ^ 8X  d`   V P                  V4      ^8  dJ   WCV P                  V4      V^,
          ,          ,          V P                  V^,
          4      ,          ,          pV# )r   y               @)r   facr   r   rj   r   r2   )r   r   rN   twopijr   s   &&&  r   polylog_continuationr     s    1us
#&&[F	
3771:Qq	&0@ AAA
AGGAJ
wwqzA~#''!*/cggajAo	CFF1I!$$SWWQqS\11Hr   c                    V P                   5pV^8  Ed(   V P                  pV P                  V4      pV P                  p^ p W,
          ^8w  dV   V P	                  W,
          4      V,          V P                  V4      ,          pV'       d   \        V4      V8  d   MWH,          pWe,          pV^,          pKv  W@P                  V4      V^,
          ,          V P                  V^,
          4      ,          V P                  V^,
          4      V P                  V P                  V4      ) 4      ,
          ,          ,          pMV^8  d   V P                  V) 4      V P                  V4      ) V^,
          ,          ,          pV P                  V4      pV P                  p	^ p
 V P                  W,
          ^,           4      pV'       dL   W,          V P                  V
4      W,
          ^,           ,          ,          p\        V4      V8  d   M!WH,          pW,          p	V
^,          p
K  \        hV P                  V4      '       d   V^ 8  d   V P                  V4      pV# r   )r   r1   r   r   rm   r   rh   harmonicr   r   rj   r   )r   r   rN   r   r   logzlogmzr   r   logkzr   r:   s   &&&         r   polylog_unitcircler     s   77(C1uHHvvayzxx}u,swwqz9CIO	MEFA	VVAY1cggacl*CLL1,=cffcffQiZ>P,PQQ	
QGGQBK#&&)qs++vvayac!e$Aw
ACE 23t9s?	MEFA
AGGAJHr   c                 "   V P                   pV P                  V4      p\        V4      ^8  g   V P                  p^V,
          pV P                  V) 4      ^V P                  ,          V,          ,          pV P                  V4      WS,          V P                  VRV,           4      ,          WS) ,          V P                  VRV,
          4      ,          ,           ,          ^V P                  ,          V,          ,          # ^p^ p V P                  W,
          4      V,          p	\        V	4      V P                  8  d   M#W9,          pV^,          pWt,          pWx,          pK]  V P                  ^V,
          4      V) V^,
          ,          ,          V,           # )   r   )r1   r   rh   r   r   gammarm   r   )
r   r(   rN   r   r   r   yr7   r   r   s
   &&&       r   polylog_generalr     s*   Aq	Aq6A:EEaCFFA2J#&&
#yy|QT#((1SU"33aeCHHQs1u<M6MMNPQRURXRXPX[\}\\	A	A
xx}q t9sww			Q		99QqS>A21+%))r   c           	      Z   V P                  V4      pV P                  V4      pV^8X  d   V P                  V4      # VR8X  d   V P                  V4      ) # V^ 8X  d   V^V,
          ,          # V^8X  d   V P                  ^V,
          4      ) # VR8X  d   V^V,
          ^,          ,          # \	        V4      R8:  g(   V P                  V4      '       g   \	        V4      R8  d   \        WV4      # \	        V4      R8  dj   V P                  V4      '       dS   RV^,           ,          \        W^V,          4      ,          \        V \        V P                  V4      4      V4      ,           # V P                  V4      '       d&   \        V \        V P                  V4      4      V4      # \        WV4      # )r         ?g?gffffff?)r   rm   altzetar   rh   r'   r   r   r&   r   r   r   )r   r(   rN   s   &&&r   polylogr     sK   AAAAAvxx{BwAAv!A#wAvqs|Bw!A#z
1v~ciills1v|ca((
1v}1ac{>#!A#669McSVWZW]W]^_W`Sacd9eee
yy||!#s366!9~q9931%%r   c                    V P                  V4      '       d(   V^ 8  d!   \        V4      ^,          ^8X  d
   V^ ,          # V'       d   V P                  V4      pMV P                  V4      pV P	                  V4      '       d8   V P	                  V4      '       d!   V P                  V P                  W4      4      # ^V,          pRV P                  W4      V P                  W4      ,
          ,          # )r   r.   )r'   r&   expjpirl   rj   imr   r   r(   rN   r   r   r:   s   &&&&  r   clsinr     s    
yy||A#a&1*/s
	JJqMHHQK
 1 1! 4 4vvckk!&''	!ACKK$s{{1'7788r   c                    V P                  V4      '       d(   V^ 8  d!   \        V4      ^,          ^ 8X  d
   V^ ,          # V'       d   V P                  V4      pMV P                  V4      pV P	                  V4      '       d8   V P	                  V4      '       d!   V P                  V P                  W4      4      # ^V,          pRV P                  W4      V P                  W4      ,           ,          # )r   r   )r'   r&   r   rl   rj   r   r   r   s   &&&&  r   clcosr     s    
yy||A#a&1*/s
	JJqMHHQK
 1 1! 4 4vvckk!&''	!AA 3;;q#3344r   c                 n     V P                   ! V3/ VB #   \         d    T P                  T4      u # i ; ir>   )_altzetark   _altzeta_generic)r   r(   rp   s   &&,r   r   r     s;    '||A((( '##A&&'s    44c                     V^8X  d   V P                   ^ V,          ,           # V P                  ^^V,
          4      ) V P                  V4      ,          # r   )ln2powm1rm   )r   r(   s   &&r   r   r   
  s@    Avww1}IIa1++r   Nc                 L   \        V4      pV^8X  d%   V'       g   V'       g    V P                  ! V3/ VB # V P                  V4      pV P                  pVP                  R4      pVP                  R4      pV'       g7   V'       g/   V P                  R4      V P                  V4      ^ ,          ,
          # V^8X  d   VR8w  d   \        V P                  V4      4      p	\        V P                  V4      4      p
\        V	4      RV,          8  d   ^
V
,          V8  d   V^8:  g   VR8X  d/     V'       d   \        R4       V P                  ! W3/ VB Wpn        # V^8X  d   V P                  # \        V4      pWP                  8X  dI   V P                  V4      V P                  8X  d    V^ 8X  d   V P                  # V P                   # V^ ,          # V P#                  V4      '       d
   ^V,          # V P                  V4      ^V P                  ,          8  d4   V^8X  d-   V'       g%   V P                  V P%                  ^V) 4      ,           # V P&                  ! WV3/ VB 5#   \         d     EL3i ; i  \         d    T'       d   \        R4        Mi ; i Ypn        ELD  Ypn        i ; i)	r   methodverboser   zeuler-maclaurinrG   zriemann-siegelz4zeta: Attempting to use the Riemann-Siegel algorithmz0zeta: Could not use the Riemann-Siegel algorithm)r&   _zetark   r   r#   rg   r   _convert_paramrh   r2   r   printrs_zetar%   r   r   r1   r   power_hurwitz)r   r(   r   r8   r   rp   r9   r#   r   r   r   absss   &&&&&,      r   rm   rm     s   JAAvqF	99Q)&)) 	AA88DZZ!Fjj#G
wws|c003A666Av&--__ r7SX"R%$,:?&&
 TU;;q??  Avwwq6Dww66!9Avww88Os
	4s

vvay1SXX:!q&ww1qb)))LLq+F+++] # 		6 + PQ 4sA   I +I. 3I. I+*I+.JJJ JJ J#c                 ~   V P                   pVP                  R 4      p ^
pV ;P                   V,          un         V P                  V4      w  r(V P                  V4      ^ 8  d(   V'       d   \	        R4        \        WW#V4      WPn         # V'       d   \	        R4        WW,           V n         \        WW#V^
,           V4      w  rV P                  V	4      V P                  W,           4      ,
          pV'       d&   \	        RV	4       \	        RV
4       \	        RVR4       W8  d   W,           WPn         # \        ^V,          \        V^,           ^dV,          4      4      pWtP                  R^dV,          4      8  g   K  V P                  R	4      h  \         d     Mi ; iT'       g   EK  \	        R4       EL*  YPn         i ; i)
r   z#zeta: Attempting reflection formulazzeta: Reflection formula failedz)zeta: Using the Euler-Maclaurin algorithmzTerm 1:zTerm 2:zCancellation:bitsmaxpreczzeta: too much cancellation)r#   rg   r   r   r   _hurwitz_reflectionrk   _hurwitz_emr   maxminNoConvergence)r   r(   r   r9   rp   r#   r   	extraprecatypeT1T2cancellations   &&&&,       r   r   r   F  sl   88Djj#G	I%%a(66!9q=;<*31?, # =>'CH tBw@FB772;7Li$i$o|V<'w 	  )S1A3t8-LM	zz)SX>>++,IJJ' ' w78$ sP   AF4 +F4 7F 
F4 BF4 0AF4 :F4 FF4 F
F4 'F4 4F<c                   a aaa V^ 8w  d   \         hS P                  V4      pV) pS P                  V4      '       d:   \        V4      pV^ 8:  d(   S P	                  ^V,
          V4      V^,
          ,          # VR8X  g   VR8X  g   \         h^V,
          o^ p^ p	VoS P                  S4      ^8  d%   S^,          oVSV,          ,          pV	^,          p	K:  S P                  S4      ^ 8:  d%   VSV,          ,          pS^,          oV	^,          p	K:   VP
                  w  p
oYS,          ,          p
^T
u;8:  d	   S8:  g   Q h Q hS P                  TT TT3R l\        ^S^,           4       4       4      pT^S P                  S4      ,          ^S P                  ,          S,          S,          ,          ,          pY,          pT#    T\        T4      8X  g   Q h\        T4      p
^o L; i)r   QZc              3      <"   T FK  pSP                  S^,          ^V,          S,          ,
          4      SP                  SVS34      ,          x  KM  	  R# 5i)r,   N)cospir   ).0r   r:   r   qr7   s   & r   	<genexpr>&_hurwitz_reflection.<locals>.<genexpr>  sH      A 1Q3qs1u9%cll1aU&;;;s   AA)
rk   r   isnpintr&   r   _mpq_fsumranger   r   )r   r(   r   r9   r   resnegsr   r   shiftprD   r:   r   r7   s   f&&&&       @@@r   r   r   k  s   Av!!
&&)C2D
{{1~~H6<<!Q'1Q3//SLESL!!	!A	AE	A
&&)a-	Q	QW

&&)q.	QW	Q
ww1
 qLA;Q;;; q1 	A399Q<366!a	''AFAHCF{{Fs   F> >!G!c           
         V P                  V4      pV) p^ pV^,          pTp	^ p
V P                  V4      '       d   \        V P                  V4      4      pV^,
          p V P	                  WV,           W,
          ^,
          V.4      ^ ,          ^ ,          pW,          p
W,           pV P                  V4      pW,          pV.p^V,          p^V,          pW) ,          pV'       d5   V P                  V^,           W,          4      W^,           ,          ,          pM^WV,          ,          ,          pVRV,          V,          ,          p^.pTp^p\        ^V	^,           4       EF  p^V,          pV^8X  d   ^.pMV^,
          V^,
          .pV F  p\        VV^,           4      pVV8:  d    VP                  VR,          V,          4       ^ .V^,           ,          p\        V4       F$  p^V,
          V,
          VV,          ,          VV&   K&  	  \        ^V^,           4       F;  pVV;;,          VV^,
          ,
          VV^,
          ,          ,          ,          uu&   K=  	  TpVV,          pK  	  V P                  VV4      V,          V P                  V4      ,          V) ,          pVV,          pV P                  V4      V8  d   V
RV,          V,          3u # VV^,           V^,           ,          ,          pEK  	  V'       d    \        RWxRV P                  X4      RV4       Y^,          rV P                  V4      ^ 8  g   EK  W^,          ,          p	EK  )r   r   z
Sum range:zterm magnitude	tolerancer   )r   r'   r&   r   _zetasumr   gammaincr  r   appendr   fdotr   r   r   r   ) r   r(   r   r9   r#   r   r   M1M2Nlsums1r   M2alogM2alogM2adlogslogrrM2aM2astailsumUr   factr   j2updsr   DUnir7   s    &&&&&&                          r   r   r     s   AA%C	
B	B
AD
yy||
O	
1B
LLqD"%'A3/215
 		d)yxuRyll1Q3	2RA#Y>G"Ri(G3=4''Cq!A#A1BAvs1bd|!A#J6KKR40S!A#YAQqSUAaDLBqE!A#A1!QqS'1QqS61A(AT	  D!A%b(99D5AAqLGwwqzCb1Ww...RTBqDM!D) * ,(8#''!*kSVWTB66!9q=AIAr   c                  a aa
a \        S P                  V4      4      RS P                  ,          8  d    S P                  VSW4V4      # S P                  VRR7      oV^ .8g  p\        V4      ^8H  pV'       g   V'       g0   S P                  VV3R l\        V^,           4       4       4      .. 3# V'       dK   V^ ,          o
S P                  VV V
V3R l\        V^,           4       4       4      pRS
,          V,          .. 3# \        V4      p	V'       g   \        V	^,           4      pV U
u. uF  p
S P                  NK  	  pp
V'       d   V U
u. uF  p
S P                  NK  	  pp
M. p\        V^,           4       EFK  pSV,           pVS,          pV'       d)   S P                  S P                  W,          ,          4      pV'       d   S P                  V4      ) pV'       dN   VV	,          pV^ ;;,          VV,          ,          uu&   V'       d   V^ ;;,          XV,          ,          uu&   K  K  S P                  pV FL  o
VS
;;,          VV,          ,          uu&   V'       d   VS
;;,          XV,          ,          uu&   VV,          pKN  	  EK  V^ ;;,          V,          uu&   V'       g   EK6  V^ ;;,          X,          uu&   EKN  	  W3#   \         d     ELi ; iu up
i u up
i )a  
Returns [xd0,xd1,...,xdr], [yd0,yd1,...ydr] where

xdk = D^k     ( 1/a^s     +  1/(a+1)^s      +  ...  +  1/(a+n)^s     )
ydk = D^k conj( 1/a^(1-s) +  1/(a+1)^(1-s)  +  ...  +  1/(a+n)^(1-s) )

D^k = kth derivative with respect to s, k ranges over the given list of
derivatives (which should consist of either a single element
or a range 0,1,...r). If reflect=False, the ydks are not computed.
r   Tr   c              3   D   <"   T F  pSV,           S,          x  K  	  R # 5ir>   rK   )r   r   r   r  s   & r   r   _zetasum.<locals>.<genexpr>  s     >+QacD[[+s    c              3      <"   T F9  pSP                  SV,           4      S,          SV,           S,          ,          x  K;  	  R # 5ir>   )r   )r   r   r   r   r9   r  s   & r   r   r%    s1     K{!!a1Q3+55{s   AAr   )rh   r   r#   _zetasum_fastrk   fnegr   r  r   r   r  r1   conjr   r   )r   r(   r   r   derivativesreflecthave_derivativeshave_one_derivativer   maxdr9   xsysr   r   xtermytermlogwr7   r  s   f&f&&&    `        @r   r
  r
    s?    366!9~chh&	$$Q17CC 88AT8"D"qc)k*a/HH>&1+>>?CCAAKvac{KKA!GaK="$${DDFm'	(Kq#((KB	( +,1chh,AaC[ET	HHSWW	23EFF1I:D"t|1%qEUT\)E  GG$AqEUQY&E1*IA	 % qEUNEw1- . 6MW # 		 
),s   K +KK$KKc           	         V P                  V4      p\        V4      p\        V4      pV^8  d   \        R4      hV P                  p V ;P                  ^
,          un        V^8X  dx   RpV FT  pV'       g   K  V^8w  g   K  RpV P
                  5p	V ;P                  ^V^,           ,          ,          un        W,          pKV  	  V'       d   V P                  5W`n        # V P                  p
\        ^V^,           4       F  pW+V,          ,          '       g   K  V^8X  d_   WW,          ,          V P                  WV3^4      V P                  WV34      V P                  V4      ,          ,
          ,          ,          p
K  WW,          ,          V P                  WV34      ,          ,          p
K  	  WV,          ,          p
W`n        V
5#   Y`n        i ; i)r,   zarbitrary order derivativesTF)r   r   r&   rk   r#   r   r%   r1   r  rm   r6   )r   r(   chir8   r   r9   r#   	have_poler   rt   rN   r  s   &&&&        r   	dirichletr7    so   AACAJA1u!"?@@88DB6I1a %IAHH1Q3'HFA  x  HHq1AQ3xx6QSSXXaA%:qE*3771:5&6 7 7A QSCHHQ1$666A  	
T	2I s,   ,G 8G A	G G 4G B%G Gc                   a aaa S P                   pVV VV3R  lpS P                  ;rgS P                  p^ p	W8  dG   Wg,          pV	^,          p	S P                  S P	                  V	4      4      oV! V	4      p\        V4      pKL  ^ p
VP                  R4      '       d   S P                  S4      pRS P                  R,          ,          \        ^V4      ,          SVR,
          ,          ,          S P                  S^S P                  ,          ,          4      ,          S P                  RSS^,          ,          4      ,          \        S P                  S^,          4      4      ,          p
\        V
4      p
V5W3# )c                 t   < SP                  R S,          SS^,          ,          RR7      SS) ,          ,          # )r   T)regularized)r  )r   r   r   gammr(   s   &r   r?   &secondzeta_main_term.<locals>.<lambda>7  s+    #,,s1uaaiT,B41":Mr   errorr   r   r/   )r   r1   r%   r   zetazero_memoizedrh   rg   r   r   r   r6   r  r   )r   r(   r   rp   r   r   totsumr   mgr   r   sgr;  s   fff,        @r   secondzeta_main_termrB  5  s   
''CMAHHF	B	A
(	Qvvc++A./tY
Czz'VVAY#&&2,s1Ry(RV4SWWT1SVV8_5MM\\$$'	*++.syy1~+>?#h7C?r   c                 4  a aa S P                   pVV V3R  lpS P                  ;rgS P                  p^p	W8  g   V	^	8  d<   Wg,          pV	^,          p	V! V	4      pV^ 8X  d   S P                  pK;  \        V4      pKH  VP	                  R4      '       d   Tp
V5X
V	3# )c                   < SP                  R ^S,
          ,          RSP                  V 4      ^,          ,          SR,          ,          4      R SP                  V 4      ,          S^,
          ,          ,          SP                  V 4      ,          SP                  V 4      ,          ^SP	                  R S,          4      ,          SP                  SP
                  4      ,          ,          # )r   r-   r   )r  r6   mangoldtr   r   r   r   r   r   r(   s   &r   r?   'secondzeta_prime_term.<locals>.<lambda>K  s    #,,sAaCycggaj!m);a"g)EF
cggaj.AaC	 ""%,,q/225((1+>	
399SU	CHHSVV,	,.r   r=  )r   r1   r%   rh   rg   )r   r(   r   rp   r   r   r?  r   r@  r   r   s   fff,       r   secondzeta_prime_termrH  I  s    
''C	.A HHF	B	A
(a!e	Qt19BTBzz'7C?r   c                 6  a aa S P                  S4      '       di   S P                  S4      ^ 8:  dS   \        \        S P                  S4      4      4      pV^,          '       g!   S P	                  R4      V) ^,          ,          # S P
                  pVV V3R lpS P                  pV! ^ 4      pS P                  p^ p	W8  d'   Wg,          pV	^,          p	V! V	4      p\        V4      pK,  SRS,          ,          V,          S P                  RS,          4      ,          p
V
# )r   z-0.25c                 z   < R S,          V ,          V RS,          ,           SP                  V 4      ,          ,          # )r-   r   )r   rF  s   &r   r?   %secondzeta_exp_term.<locals>.<lambda>c  s'    46A+#a%34r   r   )
r'   r   r&   rz   r   r   r1   r%   rh   r   )r   r(   r   r   r   r   r?  r   r@  r   r   s   fff        r   secondzeta_exp_termrL  ]  s    
yy||q	QcffQi !1uu777#qb!e,,
''C4AXXFQ4D	B	A
(	QtY	CE
6#))CE**AHr   c           	        a aa SR S^,
          ,          ,          ^S P                  S P                  4      ,          S P                  R S,          4      ,          ,          pS P                  V4      pS ;P                  V,          un        SR S^,
          ,          ,          ^S P                  S P                  4      ,          S P                  R S,          4      ,          ,          pS P
                  pVV V3R lpS P                  pS P                  p	^p
V! V
4      p\        V4      pW8  dH   W8:  dB   W,          pV
^,          p
V! V
4      pW,          pV
^,          p
V! V
4      pTp	\        V4      pKM  W,          pRS^,
          R,          ,          S P                  S P                  ^S P                  ^,          ,          S,          4      ,           S^,
          R,          ,          ,           pWMV,           ,          p^ pVP                  R4      '       d   W8  d   W8:  g   W8:  dA   S P                  ^
4      \        S P                  \        WF,          4      ^
4      4      ,          pW8  dA   S P                  ^
4      \        S P                  \        WI,          4      ^
4      4      ,          p\        VS P
                  R,          4      pS ;P                  V,          un        V5V3# )r   c                    < SP                  V R 4      ^SP                  S4      ,          V ,          ,          SP                  RV ,          4      ,          SV ,           ^,
          SP                  V 4      ,          ,          # )r   r   )r   r   r   r   rF  s   &r   r?   *secondzeta_singular_term.<locals>.<lambda>v  sX    #,,q&#((1+'99
yyQA#a%+-r   r=        ?r   r   )r   r   r   r   r#   r   r1   r%   rh   r"   r6   rg   r   r&   r   )r   r(   r   rp   factorr   r   r   r?  mg1r   r   mg2polestr   s   fff,            r   secondzeta_singular_termrV  p  s   ac^Qsxx//		#a%0@@AFIHH	Hac^Qsxx//		#a%0@@AF
''C	-AXXF
''C	AQ4D
d)C
)
	Qt	At$i
NFqsbk>399SWWR	\!^%<<qsbkIID	f	B
Czz'zggbk3swws6:r'B#CCyggbk3swws6:r'B#CC#swwrz"HH	H38Or   c           	     x   V P                  V4      pV P                  V4      pV P                  pV P                  V4      '       d   V P                  V4      ^8:  d   \	        V^,
          4      VR,          8  d   V P
                  # \        \        V P                  V4      4      4      pV^,          '       d   V P
                  # RV) ^,          ,          V P                  ^V P                  V) RR7      ,
          ^V) ^,           ,          4      ,          # V P                  p \        WV4      p\        V P                  V4      ^ 4      pV ;P                  V^,           ,          un
        \        WVRRR7      w  rp\        WVRRR7      w  rp\!        WVRR7      w  pp\        WV4      pW,           V,           pW,
          V,           V,
          pVP#                  R4      '       dK   \%        RV	4       \%        R	VR
4       \%        RV4       \%        R	VR4       \%        RV4       \%        RV4       W`n
        VP#                  R4      '       dc   \        V P                  \	        V4      4      ^ 4      p\        V^V,          ,          V P                  R,          ^V,          ,          4      pV5V3# V5#   Y`n
        i ; i)a  
Evaluates the secondary zeta function `Z(s)`, defined for
`\mathrm{Re}(s)>1` by

.. math ::

    Z(s) = \sum_{n=1}^{\infty} \frac{1}{\tau_n^s}

where `\frac12+i\tau_n` runs through the zeros of `\zeta(s)` with
imaginary part positive.

`Z(s)` extends to a meromorphic function on `\mathbb{C}`  with a
double pole at `s=1` and  simple poles at the points `-2n` for
`n=0`,  1, 2, ...

**Examples**

    >>> from mpmath import *
    >>> mp.pretty = True; mp.dps = 15
    >>> secondzeta(2)
    0.023104993115419
    >>> xi = lambda s: 0.5*s*(s-1)*pi**(-0.5*s)*gamma(0.5*s)*zeta(s)
    >>> Xi = lambda t: xi(0.5+t*j)
    >>> chop(-0.5*diff(Xi,0,n=2)/Xi(0))
    0.023104993115419

We may ask for an approximate error value::

    >>> secondzeta(0.5+100j, error=True)
    ((-0.216272011276718 - 0.844952708937228j), 2.22044604925031e-16)

The function has poles at the negative odd integers,
and dyadic rational values at the negative even integers::

    >>> mp.dps = 30
    >>> secondzeta(-8)
    -0.67236328125
    >>> secondzeta(-7)
    +inf

**Implementation notes**

The function is computed as sum of four terms `Z(s)=A(s)-P(s)+E(s)-S(s)`
respectively main, prime, exponential and singular terms.
The main term `A(s)` is computed from the zeros of zeta.
The prime term depends on the von Mangoldt function.
The singular term is responsible for the poles of the function.

The four terms depends on a small parameter `a`. We may change the
value of `a`. Theoretically this has no effect on the sum of the four
terms, but in practice may be important.

A smaller value of the parameter `a` makes `A(s)` depend on
a smaller number of zeros of zeta, but `P(s)`  uses more values of
von Mangoldt function.

We may also add a verbose option to obtain data about the
values of the four terms.

    >>> mp.dps = 10
    >>> secondzeta(0.5 + 40j, error=True, verbose=True)
    main term = (-30190318549.138656312556 - 13964804384.624622876523j)
        computed using 19 zeros of zeta
    prime term = (132717176.89212754625045 + 188980555.17563978290601j)
        computed using 9 values of the von Mangoldt function
    exponential term = (542447428666.07179812536 + 362434922978.80192435203j)
    singular term = (512124392939.98154322355 + 348281138038.65531023921j)
    ((0.059471043 + 0.3463514534j), 1.455191523e-11)

    >>> secondzeta(0.5 + 40j, a=0.04, error=True, verbose=True)
    main term = (-151962888.19606243907725 - 217930683.90210294051982j)
        computed using 9 zeros of zeta
    prime term = (2476659342.3038722372461 + 28711581821.921627163136j)
        computed using 37 values of the von Mangoldt function
    exponential term = (178506047114.7838188264 + 819674143244.45677330576j)
    singular term = (175877424884.22441310708 + 790744630738.28669174871j)
    ((0.059471043 + 0.3463514534j), 1.455191523e-11)

Notice the great cancellation between the four terms. Changing `a`, the
four terms are very different numbers but the cancellation gives
the good value of Z(s).

**References**

A. Voros, Zeta functions for the Riemann zeros, Ann. Institute Fourier,
53, (2003) 665--699.

A. Voros, Zeta functions over Zeros of Zeta Functions, Lecture Notes
of the Unione Matematica Italiana, Springer, 2009.
r   Tr#  True)r=  r   )r=  r   zmain term =z    computed usingzzeros of zetazprime term =z#values of the von Mangoldt functionzexponential term =zsingular term =r=  rP  r   )r   r   r'   r   rh   r%   r&   rz   fractionr   r#   rL  r   r   rB  rH  rV  rg   r   )r   r(   r   rp   r   r   r#   t3r   rq   r1gtrr   r2ptt4r4r   r7   r   s   &&&,                r   
secondzetara    s-   x 	AAAA
''C
yy||q	Qqs8c$h77NcffQi !q5577NA2q5M<<#,,r,"= =a1"Q$iHI J88D +A&	IM!)#O
*3P
)#?B +eBhE"HRK::i  -$&O<."%&,QR&+#R(zz'A"#ad(CGGBJq!tO,r3w2I s   !DJ1 1J9c                  a aaa	a
a V^ 8X  d   SS) ,          # V^8X  d   S P                  SS4      # S^8X  d   S P                  SV4      V,          # S P                  S4      ^8  d   S P                  S4      '       d   \	        R4      h\        S P                  ^S P                  S4      ,
          4      4      pS P                  pS P                  p\        V4       F(  pWVSV,           S,          ,          ,          pWa,          pK*  	  VS P                  VSSV,           4      ,          V,           # S P                  V4      o	^^SS,          ,          ,          S P                  ^S,
          S) S	,          4      S	) S^,
          ,          ,          VS,          ,          ,           pS^,          o
^S P                  ,          oVV V	V
VV3R lpV^S P                  V^ S P                  .4      ,          ,          pS P!                  V4      '       gV   S P!                  S4      '       g?   S P!                  S4      '       g(   S P                  V4      ^8  d   S P#                  V4      pV# )a	  
Gives the Lerch transcendent, defined for `|z| < 1` and
`\Re{a} > 0` by

.. math ::

    \Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(a+k)^s}

and generally by the recurrence `\Phi(z,s,a) = z \Phi(z,s,a+1) + a^{-s}`
along with the integral representation valid for `\Re{a} > 0`

.. math ::

    \Phi(z,s,a) = \frac{1}{2 a^s} +
            \int_0^{\infty} \frac{z^t}{(a+t)^s} dt -
            2 \int_0^{\infty} \frac{\sin(t \log z - s
                \operatorname{arctan}(t/a)}{(a^2 + t^2)^{s/2}
                (e^{2 \pi t}-1)} dt.

The Lerch transcendent generalizes the Hurwitz zeta function :func:`zeta`
(`z = 1`) and the polylogarithm :func:`polylog` (`a = 1`).

**Examples**

Several evaluations in terms of simpler functions::

    >>> from mpmath import *
    >>> mp.dps = 25; mp.pretty = True
    >>> lerchphi(-1,2,0.5); 4*catalan
    3.663862376708876060218414
    3.663862376708876060218414
    >>> diff(lerchphi, (-1,-2,1), (0,1,0)); 7*zeta(3)/(4*pi**2)
    0.2131391994087528954617607
    0.2131391994087528954617607
    >>> lerchphi(-4,1,1); log(5)/4
    0.4023594781085250936501898
    0.4023594781085250936501898
    >>> lerchphi(-3+2j,1,0.5); 2*atanh(sqrt(-3+2j))/sqrt(-3+2j)
    (1.142423447120257137774002 + 0.2118232380980201350495795j)
    (1.142423447120257137774002 + 0.2118232380980201350495795j)

Evaluation works for complex arguments and `|z| \ge 1`::

    >>> lerchphi(1+2j, 3-j, 4+2j)
    (0.002025009957009908600539469 + 0.003327897536813558807438089j)
    >>> lerchphi(-2,2,-2.5)
    -12.28676272353094275265944
    >>> lerchphi(10,10,10)
    (-4.462130727102185701817349e-11 - 1.575172198981096218823481e-12j)
    >>> lerchphi(10,10,-10.5)
    (112658784011940.5605789002 - 498113185.5756221777743631j)

Some degenerate cases::

    >>> lerchphi(0,1,2)
    0.5
    >>> lerchphi(0,1,-2)
    -0.5

Reduction to simpler functions::

    >>> lerchphi(1, 4.25+1j, 1)
    (1.044674457556746668033975 - 0.04674508654012658932271226j)
    >>> zeta(4.25+1j)
    (1.044674457556746668033975 - 0.04674508654012658932271226j)
    >>> lerchphi(1 - 0.5**10, 4.25+1j, 1)
    (1.044629338021507546737197 - 0.04667768813963388181708101j)
    >>> lerchphi(3, 4, 1)
    (1.249503297023366545192592 - 0.2314252413375664776474462j)
    >>> polylog(4, 3) / 3
    (1.249503297023366545192592 - 0.2314252413375664776474462j)
    >>> lerchphi(3, 4, 1 - 0.5**10)
    (1.253978063946663945672674 - 0.2316736622836535468765376j)

**References**

1. [DLMF]_ section 25.14

z#Lerch transcendent complex infinityc                    < SP                  SSP                  V S,          4      ,          V S,          ,
          4      S^,          V ^,          ,           S,          SP                  SV ,          4      ,          ,          # rJ   )sinatanexpm1)r7   r   r   rD   rt   r   r(   s   &r   r?   lerchphi.<locals>.<lambda>}  sP    #''!CHHQqSM/!A#-.
Q$q!t)a#))AaC.	(*r   )rm   r   r   r   r   r&   r   r1   r   r   lerchphir   r  r   r$   r%   r   chop)r   rN   r(   r   r   r   zpowr   r   rD   rt   r   s   f&ff     @@@r   rh  rh    s   b 	AvaRyAvxx1~Av{{1a 1$$
vvay1};;q>>BCC366!9%&HHwwA1q AID  cll1Q!,,q00q	A	1QT6
S\\!A#r!t,ac{:QTAAA	AA	#&&A	* 	*A388A377|$	$$A66!99SVVAYYsvvayySVVAY]HHQKHr   r   )r   )z2http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros1)F)r   r   N)r   r   )gQ?)+
__future__r   libmp.backendr   	functionsr   r   r   r*   r;   rE   re   r{   r~   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   rm   r   r   r   r
  r7  rB  rH  rL  rV  ra  rh  rK   r   r   <module>rn     s&   % " 9 9# #J + +B B B J% J%Z &'34@ &'34@   ' (4 5A 	 	 '	 (4	 5A	
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E 
E  , # #   % %F *) *)X %) %)N . .	"H*( & &* 
9 
9 
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5 ' ' , ,
 3, 3,j " "H'R=B ()sE ; ;z #!  B((&"H  D j jr   