[fjvphlki] Counterexamples to GRH

People have searched really hard for counterexamples to the Riemann Hypothesis, zeroes off the critical line of the Riemann zeta function.  They have searched less hard for counterexamples to the Generalized Riemann Hypothesis, zeroes off the critical line in any one of the infinite number of Dirichlet L-functions.  A huge number of conditional proofs rely on GRH being true (probably a greater number than those that rely on the regular RH being true), so finding a counterexample to GRH would be very important (though would not necessarily invalidate any particular conditional proof).

Can we construct L-functions that are heuristically more likely to have counterexamples, functions which should be searched first?  Perhaps those which are least similar to the Riemann zeta function by some metric.