L-functions as distributions

We define an axiomatic class of -functions extending the Selberg class. We show in particular that one can recast the traditional conditions of an Euler product, analytic continuation and functional equation in terms of distributional identities akin to Weil's explicit formula. The generality of our approach enables some new applications; for instance, we show that the -function of any cuspidal automorphic representation of has infinitely many zeros of odd order.