  A new formula relating the analytic continuation of the Hurwitz zeta function
to the Euler gamma function and a polylogarithmic function is presented. In
particular, the values of the first derivative of the real part of the analytic
continuation of the Hurwitz zeta function for even negative integers and the
imaginary one for odd negative integers are explicitly given. The result can be
of interest both on mathematical and physical side, because we are able to
apply our new formulas in the context of the Spectral Zeta Function
regularization, computing the exact pair production rate per space-time unit of
massive Dirac particles interacting with a purely electric background field.
