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: let $k/2$ be the center of the critical strip, $d$ be the
 degree, $\kbd{Vga} = (\alpha_{j})_{j\leq d}$ given the gamma factors,
 and $\varepsilon$ be the root number; we set
 $s = k/2+it = \rho e^{i\theta}$ and
 $2E = d(k/2-1) + \Re(\sum_{1\le j\le d}\alpha_{j})$. Assume first that
 $\Lambda$ is self-dual, then the computed function at $t$ is equal to
 $$Z(t) = \varepsilon^{-1/2}\Lambda(s) \cdot \rho^{-E}e^{dt\theta/2}\;,$$
 which is a real function of $t$
 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
 $Z(t) \approx 1$. For non-self-dual $\Lambda$, the definition is the same
 except we drop the $\varepsilon^{-1/2}$ term (which is not well defined since
 it depends on the chosen dual sequence $a^{*}(n)$): $Z(t)$ is still of the
 order of $1$ and still vanishes where $L(k/2+it)$ does, but it needs no
 longer be real-valued.

 \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.
