L-functions of Maass forms (reviewed)

In 1949 Hans Maass made his profound discovery that there are L-functions associated with non-holomorphic automorphic forms. In his Math Review of Maass' article [MR:0031519] J. Lehner writes

In Hecke's theory of Dirichlet series with Euler products we associate, roughly speaking, a Dirichlet series with an automorphic function; the invariance of the latter under linear substitutions is used, together with the Mellin transform, to derive a functional equation for the Dirichlet series. This suffices for the discussion of the $\zeta$-function of an imaginary quadratic field, for example, but not of a real quadratic field. In order to handle the latter case, the author defines a class of functions ("automorphic wave functions") to take the place of the analytic automorphic functions of Hecke's theory.

Maass' work prompted André Weil to remark “Il a fallu Maass pour nous sortir du ghetto des fonctions holomorphes.”