On a version of Hadamard's method in the theory of Dirichlet's $L$-functions

In the paper a new version of the Hadamard's method in the theory of Dirichlet's $L$-functions is given. We prove of this method of the absence of the $L$-functions zeroes on the unit line. We show that the Hadamard's method allow to get results, which on the accuracy correspond to the Vallee Poussin results in the asymptotical law of the distribution of primes. Of this we extend possibilities of the Hadamard's method. New estimations of the zeta-sum twisted together with the Dirichlet's character by modulo, equals to the degree of an odd prime number are obtained that permits to get the modern limit of zeroes for the corresponding Dirichlet's $L$-function.