  Let $L(s,\chi)$ be the Dirichlet $L$-function associated to a non trivial
primitive Dirichlet character $\chi$ defined $\bmod\ q$, where $q$ is an odd
prime. In this paper we introduce a fast method to compute $\vert L(1,\chi)
\vert$ using the values of Euler's $\Gamma$ function. We also introduce an
alternative way of computing $\log \Gamma(x)$ and $\psi(x)=
\Gamma^\prime/\Gamma(x)$,$x\in(0,1)$. Using such algorithms we numerically
verify the classical Littlewood bounds and the recent Lamzouri-Li-Soundararajan
estimates on $\vert L(1,\chi) \vert$, where $\chi$ runs over the non trivial
primitive Dirichlet characters $\bmod\ q$, for every odd prime $q$ up to
$10^7$. The programs used and the results here described are collected at the
following address
\url{http://www.math.unipd.it/~languasc/Littlewood_ineq.html}.
