lfun

Function: lfun
Section: l_functions
C-Name: lfun0
Prototype: GGD0,L,b
Help: lfun(L,s,{D=0}): compute the L-function value L(s), or
 if D is set, the derivative of order D at s. L is either an
 Lmath, an Ldata or an Linit.
Description:
 (gen,gen):gen:prec       lfun($1, $2, $bitprec)
 (gen,gen,?0):gen:prec    lfun($1, $2, $bitprec)
 (gen,gen,small):gen:prec lfun0($1, $2, $3, $bitprec)
Doc: compute the L-function value $L(s)$, or if \kbd{D} is set, the
 derivative of order \kbd{D} at $s$. The parameter
 \kbd{L} is either an Lmath, an Ldata (created by \kbd{lfuncreate}, or an
 Linit (created by \kbd{lfuninit}), preferrably the latter if many values
 are to be computed.

 The argument $s$ is also allowed to be a power series; for instance, if $s =
 \alpha + x + O(x^n)$, the function returns the Taylor expansion of order $n$
 around $\alpha$. The result is given with absolute error less than $2^{-B}$,
 where $B = \text{realbitprecision}$.

 \misctitle{Caveat} The requested precision has a major impact on runtimes.
 It is advised to manipulate precision via \tet{realbitprecision} as
  explained above instead of \tet{realprecision} as the latter allows less
 granularity: \kbd{realprecision} increases by increments of 64 bits, i.e. 19
 decimal digits at a time.

 \bprog
 ? lfun(x^2+1, 2)  \\ Lmath: Dedekind zeta for Q(i) at 2
 %1 = 1.5067030099229850308865650481820713960

 ? L = lfuncreate(ellinit("5077a1")); \\ Ldata: Hasse-Weil zeta function
 ? lfun(L, 1+x+O(x^4))  \\ zero of order 3 at the central point
 %3 = 0.E-58 - 5.[...] E-40*x + 9.[...] E-40*x^2 + 1.7318[...]*x^3 + O(x^4)

 \\ Linit: zeta(1/2+it), |t| < 100, and derivative
 ? L = lfuninit(1, [100], 1);
 ? T = lfunzeros(L, [1,25]);
 %5 = [14.134725[...], 21.022039[...]]
 ? z = 1/2 + I*T[1];
 ? abs( lfun(L, z) )
 %7 = 8.7066865533412207420780392991125136196 E-39
 ? abs( lfun(L, z, 1) )
 %8 = 0.79316043335650611601389756527435211412  \\ simple zero
 @eprog