Numerical computation of Maass forms and associated L-functions

In order to extend the theory of Dirichlet series with Euler products,
Maass studied non-holomorphic automorphic functions, nowadays called Maass
forms.  While the existence of Maass cusp forms is known by the
Roelcke-Selberg spectral resolution, their explicit form can only be
approximated numerically.  We recall the yet most successful algorithm for
approximating Maass forms and explore some of their properties.  Then, we
present an Odlyzko-Schoenhage type algorithm which computes Maass
L-functions high up in the critical strip.