  Using some basic properties of the gamma function, we evaluate a simple class
of infinite products involving Dirichlet characters as a finite product of
gamma functions and, in the case of odd characters, as a finite product of
sines. As a consequence we obtain evaluations of certain multiple $L$-series.
In the final part of this paper we derive expressions for infinite products of
cyclotomic polynomials, again as finite products of gamma or of sine functions.
