  In this paper, we show that the regularized determinants of some Dirichlet
series are multiplicative. As an application, we give generalizations of
Lerch's formula for the classical gamma function and we determine the sum of
some Dirichlet series generalizing Euler's formula on the sum of the reciprocal
of squares. We recover the results of Kurokawa and Wakayama, and give a new
proof for some Euler's formulas.
