New theoretical and numerical investigations of the following two problems associated with the Dirichlet L-functions L(s; χ) are carried out in this work: the conjecture on the values L(1/2; χ) and the Extended Riemann Hypothesis for the function L(s; χ) with a character χ = χ(n) being equal to a Legendre symbol (n/p), 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.

Mathematical problems and theory of numerical methods