[ltsqarmj] engineering a GRH counterexample

to find a counterexample to the Riemann Hypothesis, thereby disproving it, you need to find a weird zero (root) of the Riemann zeta function.

to find a counterexample to the Generalized Riemann Hypothesis or Extended Riemann Hypothesis, you can choose any L-function you want and find a weird zero of it.  (L-functions are a set of functions that includes the Riemann zeta function.)  this flexibility seems to make hunting for counterexamples to GRH or ERH more attractive than original RH.  a disproof of GRH or ERH would be as mathematically significant as disproving RH.  what are some techniques that might help to construct L-functions favorable for finding counterexamples?

or, you could also construct a special L-function and prove GRH or ERH on just your one function, not proving the hypotheses in general.  has this already been done?  maybe there are some easy cases, like L-functions which provably have no nontrivial roots at all, so "all" the nontrivial roots have any property you wish (vacuous truth).

vaguely similar reasoning lead to the proof of Fermat's Last Theorem.  trying to construct an elliptic curve with properties favorable for leading to a counterexample to FLT lead to a proof that no such elliptic curve exists.