Function: lfunhardy
Section: l_functions
C-Name: lfunhardy
Prototype: GGb
Help: lfunhardy(L,t): variant of the Hardy L-function attached to L, used for
 plotting on the critical line.
Doc: Variant of the Hardy $Z$-function given by \kbd{L}, used for
 plotting or locating zeros of $L(k/2+it)$ on the critical line.
 The precise definition is as
 follows: if as usual $k/2$ is the center of the critical strip, $d$ is the
 degree, $\alpha_j$ the entries of \kbd{Vga} giving the gamma factors,
 and $\varepsilon$ the root number, then if we set
 $s = k/2+it = \rho e^{i\theta}$ and
 $E=(d(k/2-1)+\sum_{1\le j\le d}\alpha_j)/2$, the computed function at $t$ is
 equal to
 $$Z(t) = \varepsilon^{-1/2}\Lambda(s) \cdot |s|^{-E}e^{dt\theta/2}\;,$$
 which is a real function of $t$ for self-dual $\Lambda$,
 vanishing exactly when $L(k/2+it)$ does on the critical line. The
 normalizing factor $|s|^{-E}e^{dt\theta/2}$ compensates the
 exponential decrease of $\gamma_A(s)$ as $t\to\infty$ so that
 $Z(t) \approx 1$.

 \bprog
 ? T = 100; \\ maximal height
 ? L = lfuninit(1, [T]); \\ initialize for zeta(1/2+it), |t|<T
 ? \p19 \\ no need for large accuracy
 ? ploth(t = 0, T, lfunhardy(L,t))
 @eprog\noindent Using \kbd{lfuninit} is critical for this particular
 applications since thousands of values are computed. Make sure to initialize
 up to the maximal $t$ needed: otherwise expect to see many warnings for
 unsufficient initialization and suffer major slowdowns.