lfuninit

Function: lfuninit
Section: l_functions
C-Name: lfuninit0
Prototype: GGD0,L,b
Help: lfuninit(L,sdom,{der=0}): precompute data
 for evaluating the L-function given by 'L' (and its derivatives
 of order der, if set) in rectangular domain sdom = [center,w,h]
 centered on the real axis, |Re(s)-center| <= w, |Im(s)| <= h,
 where all three components of sdom are real and w,h are nonnegative.
 The subdomain [k/2, 0, h] on the critical line can be encoded as [h] for
 brevity.
Doc: initalization function for all functions linked to the
 computation of the $L$-function $L(s)$ encoded by \kbd{L}, where
 $s$ belongs to the rectangular domain $\kbd{sdom} = [\var{center},w,h]$
 centered on the real axis, $|\Re(s)-\var{center}| \leq w$, $|\Im(s)| \leq h$,
 where all three components of \kbd{sdom} are real and $w$, $h$ are
 nonnegative. \kbd{der} is the maximum order of derivation that will be used.
 The subdomain $[k/2, 0, h]$ on the critical line (up to height $h$)
 can be encoded as $[h]$ for brevity. The subdomain $[k/2, w, h]$
 centered on the critical line can be encoded as $[w, h]$ for brevity.

 The argument \kbd{L} is an \kbd{Lmath}, an \kbd{Ldata} or an \kbd{Linit}. See
 \kbd{??Ldata} and \kbd{??lfuncreate} for how to create it.

 The height $h$ of the domain is a \emph{crucial} parameter: if you only
 need $L(s)$ for real $s$, set $h$ to~0.
 The running time is roughly proportional to
 $$(B / d+\pi h/4)^{d/2+3}N^{1/2},$$
 where $B$ is the default bit accuracy, $d$ is the degree of the
 $L$-function, and $N$ is the conductor (the exponent $d/2+3$ is reduced
 to $d/2+2$ when $d=1$ and $d=2$). There is also a dependency on $w$,
 which is less crucial, but make sure to use the smallest rectangular
 domain that you need.
 \bprog
 ? L0 = lfuncreate(1); \\ Riemann zeta
 ? L = lfuninit(L0, [1/2, 0, 100]); \\ for zeta(1/2+it), |t| < 100
 ? lfun(L, 1/2 + I)
 ? L = lfuninit(L0, [100]); \\ same as above !
 @eprog

Function: _lfuninit_worker
C-Name: lfuninit_worker
Prototype: LGGGGGGGG
Section: programming/internals
Help: worker for lfuninit

Function: _lfuninit_theta2_worker
C-Name: lfuninit_theta2_worker
Prototype: LGGGGGG
Section: programming/internals
Help: worker for lfuninit using theta2