The Laplace transform of Dirichlet $L$-functions

Let $\chi$ be a Dirichlet character modulo $q$. The Dirichlet $L$-function $L(s,\chi)$ is defined in the half-plane $\sigma>1$ by the series
$$ L(s,\chi)=\sum_{m=1}^{\infty}\frac{\chi(m)}{m^s}, $$
and has a meromorphic continuation to the whole complex plane. If $\chi$ is a non-principal character, then the function $L(s,\chi)$ is entire one. In the case of the principal character, the function $L(s,\chi)$ has unique simple pole at the point $s=1$. Dirichlet $L$- functions play an important role in the investigations of the distribution of prime numbers in arithmetical progresions, therefore, their analytic properties deserve a constant attention. In applications, often the moments of Dirichlet $L$-functions are used, whose asymptotic behaviour is very complicated. For investigation of moments, various methods are applied, one of them is based on the application of Mellin transforms. On the other hand, Mellin transforms use Laplace transforms. In the paper, the formulae for the Laplace transform of the function $\arrowvert L(s,\chi) \arrowvert^2$ in the critical strip are obtained. They extend the formulae obtained in [BaLa] on the critical line $\sigma=\frac{1}{2}$.