  We present a group-theoretic criterion under which one may verify the Artin
conjecture for some (non-monomial) Galois representations, up to finite height
in the complex plane. In particular, the criterion applies to S5 and A5
representations. Under more general conditions, the technique allows for the
possibility of verifying the Riemann hypothesis for Dedekind zeta functions of
non-abelian extensions of Q. In addition, we discuss two methods for locating
zeros of arbitrary L-functions. The first uses the explicit formula and
techniques (developed jointly with Andreas Strombergsson) for computing with
trace formulae. The second method generalizes that of Turing for verifying the
Riemann hypothesis. In order to apply it we develop a rigorous algorithm for
computing general L-functions on the critical line via the Fast Fourier
Transform. Finally, we present some numerical results testing Artin's
conjecture for S5 representations, and the Riemann hypothesis for Dedekind zeta
functions of S5 and A5 fields.
