One of the important properties of the Dirichlet L-functions is that they may be expanded as Euler products. Unfortunately there is no corresponding expansion for the Kubota-Leopoldt p-adic L-functions since the infinite products involved do not converge. In this paper we give an explicit formula which shows how any finite number of Euler factors may be factored off in a natural manner. Of course, Euler factors may be factored from any function; but we show that, among p-adic analogues of Dirichlet series with periodic coefficients, the p-adic L-functions are precisely the ones which allow this to be done in a natural way which we call a weak Euler product. This might indicate that weak Euler products are the best that can be hoped for. Finally we make a few remarks concerning other analytic properties of the p-adic L-functions.