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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?

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 @Marius: Can you give an example to the following: "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." – GH Aug 9 2011 at 14:42
@GH: Actually, I was quoting Atle Selberg from a long interview that he gave about a year before he died. I have to scratch around for the interview, but suspect that he was thinking of the Selberg zeta function for a generic compact Riemann surface. I will report back. – Marius Overholt Aug 9 2011 at 14:55
(There are also dynamical zeta functions: ams.org/notices/200208/fea-ruelle.pdf ) – Qfwfq Aug 9 2011 at 15:05
@Marius: Thanks, that makes sense. – GH Aug 9 2011 at 18:49
@GH: It was the way I thought I remembered it. Selberg made the remark in part 3 of the interview, about the Riemann Hypothesis and the trace formula. The interviewers were Nils A. Baas and Christian F. Skau. I translate their question from Norwegian into English: Have you considered whether there exists any kind of geometrical analogies to the primes in a fundamental sense? Selberg's answer translated from the Norwegian: – Marius Overholt Aug 10 2011 at 6:03
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