Axioms for zeta functions 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?
 A: It is not true that the Selberg class includes only L-functions that "come from arithemtic". In fact, all the "reasonable geometric L-functions", that is, motivic L-functions are expected to belong to it.
For an unconditional example, the Hasse-Weil L-function of elliptic curves over $\mathbb{Q}$ are known to be in the Selberg class (ultimately this follows from the modularity theorem of Breuil-Conrad-Taylor-Wiles).
The only class of L-functions more general than Selberg's that (as far as I know) has been studied in some depth is the extended Selberg class $S♯$, 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 of the axioms is that it allows L-functions that behave too badly for any reasonable version of the 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, and restricted after that if necessary (Kaczorowski & Perelli classification results are directly on the extended case, for example).
A: Andrew Booker defined the notion of L-datum that encompasses the Selberg class and allows to obtain simplicity results for the non trivial zeroes of some automorphic L-functions. Thomas Oliver, building on his work, began to classify L-data of low degrees.
