Mellin integral representation of Dirichlet series

Question

Are there any known representations of $D(s)$ a Dirichlet series, or let's say a Dirichlet L-function, of the type \begin{equation} D(s)=\int_0^\infty f(x)\,\mathrm{d}x \end{equation} with $f:[0,\infty)\longrightarrow \mathbb{C}$? Like we have for the Zeta function, \begin{equation} \zeta(s)=\int_0^\infty \frac{1}{\Gamma(s)}\frac{x^{s-1}}{e^x-1}\,\mathrm{d}x \end{equation} for $\sigma>1$.

Answer 1

$$\sum_{n=1}^\infty a_n n^{-s} = s \int_1^\infty (\sum_{n \le x} a_n) x^{-s-1}dx = \frac{1}{\Gamma(s)} \int_0^\infty (\sum_{n=1}^\infty a_n e^{-nx}) x^{s-1}dx$$

To understand what it means in full generality, you should consider the Laplace/Fourier transform of the distribution $\sum_{n=1}^\infty a_nn^{-\sigma} \delta(u-\log n)$ and apply to it the convolution theorems.

The Perron formula is then a special case of the Laplace/Fourier inversion theorem.