  We study $p$-adic $L$-functions $L_p(s,\chi)$ for Dirichlet characters
$\chi$. We show that $L_p(s,\chi)$ has a Dirichlet series expansion for each
regularization parameter $c$ that is prime to $p$ and the conductor of $\chi$.
The expansion is proved by transforming a known formula for $p$-adic
$L$-functions and by controlling the limiting behavior. A finite number of
Euler factors can be factored off in a natural manner from the $p$-adic
Dirichlet series. We also provide an alternative proof of the expansion using
$p$-adic measures and give an explicit formula for the values of the
regularized Bernoulli distribution. The result is particularly simple for
$c=2$, where we obtain a Dirichlet series expansion that is similar to the
complex case.
