  Let $q$ be a prime, $\chi$ be a non-principal Dirichlet character $\bmod\ q$
and $L(s,\chi)$ be the associated Dirichlet $L$-function. For every odd prime
$q\le 10^7$, we show that $L(1,\chi_\square) > c_{1} \log q$ and $\beta < 1-
\frac{c_{2}}{\log q}$, where $c_1=0.0124862668\dotsc$,
$c_2=0.0091904477\dotsc$, $\chi_{\square}$ is the quadratic Dirichlet character
$\bmod\ q$ and $\beta\in (0,1)$ is the Landau-Siegel zero, if it exists, of
such a set of Dirichlet $L$-functions. As a by-product of the computations here
performed, we also obtained some information about the Littlewood and Joshi
bounds on $L(1,\chi_\square)$ and on the class number of the imaginary
quadratic field ${\mathbb Q}(\sqrt{-q})$.
