The connection between the values of the L-functions at the point s=1/2 and Gauss sums which leads to the sufficient condition for the validity of the equality L(1/2,χ)=0 is studied. The necessary conditions for the validity of the Extended Riemann Hypothesis for the L-fuctions are given in terms of the signs of the even-order derivatives of the fuctionn ξ(s,χ) which is an analogue of the Riemann ξ-function ξ(s). All the results are applied to the L-functions L(s,χ)$ with a character χ being equal to a Legendre symbol.

Mathematical problems and theory of numerical methods