  Branched covering Riemann surfaces $(\mathbb{C},f)$ are studied, where $f$ is
the Euler Gamma function and the Riemann Zeta function. For both of them
fundamental domains are found and the group of covering transformations is
revealed. In order to find fundamental domains, pre-images of the real axis are
taken and a thorough study of their geometry is performed. The technique of
simultaneous continuation, introduced by the authors in previous papers, is
used for this purpose. Color visualization of the conformal mapping of the
complex plane by these functions is used for a better understanding of the
theory. For the Riemann Zeta function the outstanding question of the
multiplicity of its zeros, as well as of the zeros of its derivative is
answered.
