The Selberg class is an axiomatization of arithmetically significant zeta functions (a.k.a. Lfunctions) 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?

The only class of Lfunctions more general than Selberg's that (as far as I know) has been studied at some depth is the extended Selberg class, which removes the Ramanujan and Euler product restrictions (see, for example Kaczorowski & Perelli (2002) On the structure of the Selberg class, V) The problem with any further extension on the axioms is that any relaxation seems to allow Lfunctions that behave too bad for any reasonable modified Riemann hypothesis to hold. In fact, the reason that makes extended Selberg class interesting is that anything that so far has been proved about Selberg class can be proved in the extended setting too (Kaczorowski & Perelli classification results are directly on the extended case, for example). 

