Numerical estimates on the Landau-Siegel zero and other related quantities

Let $q$ be a prime, $\chi$ be a non-principal Dirichlet character $q$ and $L(s,\chi )$ be the associated Dirichlet $L$-function. For every odd prime $q\le {10}^7$, we show that $L(1,{\chi }_{\mathrm{\square }})>c_1\log q$ and $\beta <1-\frac{c_2}{\log q}$, where $c_1=0.0124862668\dots$, $c_2=0.0091904477\dots$, ${\chi }_{\mathrm{\square }}$ is the quadratic Dirichlet character $q$ 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 $L(1,{\chi }_{\mathrm{\square }})$ and on the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$.