New results on the Dirichlet $L$-functions

New theoretical and numerical investigations of the following two problems associated with the Dirichlet $L$-functions $L(s,\xi)$ are carried out in this work: the conjecture on the values $L(1/2,\xi)$ and the Extended Riemann Hypothesis for the function $L(s,\xi)$ with a character $\xi=\xi(n)$ being equal to a Legendre symbol $\bigl(\frac np\bigr)$, where $p$ is a prime. New rigorous theoretical results give necessary and sufficient conditions for the validity or refutation of the second conjecture. Numerical investigations performed with a computer for all $p<500000$ confirm the necessary condition and do not confirm conditions sufficient to its refutation. An analytic approximation of the numerical distribution is found as well.