Conjecturally, they include all examples of L-functions, and in particular are expected to coincide with the Selberg class.
Furthermore, all L-functions over arbitrary number fields are widely thought to be instances of standard L-functions for the general linear group GL(n) over the rational numbers Q.
This makes them a useful testing ground for statements about L-functions, since it sometimes affords structure from the theory of automorphic forms.
These L-functions were proven to always be entire by Roger Godement and Hervé Jacquet,[3] with the sole exception of Riemann ζ-function, which arises for n = 1.
Another proof was later given by Freydoon Shahidi using the Langlands–Shahidi method.