The Selberg class is an axiomatization of arithmetically significant zeta functions (a.k.a. L-functions) by a few analytic properties (functional equation etc.) However there do exist other zeta functions that do not seem to come from arithmetic, but instead from geometry, for example. Some of these are known to have zeros on the real axis to the right of the critical line, but are otherwise expected to satisfy an analogue of the Riemann Hypothesis.
Have any attempt been made to write down axioms for such more general zeta functions that are expected to satisfy an appropriately modified Riemann Hypothesis?