  After recalling the precise existence conditions of the zeta function of a
pseudodifferential operator, and the concept of reflection formula, an
exponentially convergent expression for the analytic continuation of a
multidimensional inhomogeneous Epstein-type zeta function of the general form
\zeta_{A,\vec{b},q} (s) = \sum_{\vec{n}\in Z^p (\vec{n}^T A \vec{n} +\vec{b}^T
\vec{n}+q)^{-s}, with $A$ the $p\times p$ matrix of a quadratic form, $\vec{b}$
a $p$ vector and $q$ a constant, is obtained. It is valid on the whole complex
$s$-plane, is exponentially convergent and provides the residua at the poles
explicitly. It reduces to the famous formula of Chowla and Selberg in the
particular case $p=2$, $\vec{b}= \vec{0}$, $q=0$. Some variations of the
formula and physical applications are considered.
