  When it comes to partial numerical verification of the Riemann Hypothesis,
one crucial part is to verify the completeness of a list of pre-computed zeros.
Turing developed such a method, based on an explicit version of a theorem of
Littlewood on the average of the argument of the Riemann zeta function. In a
previous paper we suggested an alternative method based on the Weil-Barner
explicit formula. This method asymptotically sacrifices fewer zeros in order to
prove the completeness of a list of zeros with imaginary part in a given
interval. In this paper, we prove a general version of this method for an
extension of the Selberg class including Hecke and Artin L-series, L-functions
of modular forms, and, at least in the unramified case, automorphic
L-functions. As an example, we further specify this method for Hecke L-series
and L-functions of elliptic curves over the rational numbers.
