  We dealt in a series of previous publications with some geometric aspects of
the mappings by functions obtained as analytic continuations to the whole
complex plane of general Dirichlet series. Pictures illustrating those aspects
contain a lot of other information which has been waiting for a rigorous proof.
Such a task is partially fulfilled in this paper, where we succeeded among
other things, to prove a theorem about general Dirichlet series having as
corollary the Speiser's theorem. We have also proved that those functions do
not possess multiple zeros of order higher than $2$ and the double zeros have
very particular locations. Moreover, their derivatives have only simple zeros.
With these results at hand, we revisited GRH for a simplified proof.
