One metric result about analytic continuation of some Dirichlet series

In this paper we consider certain 1-parametric family of Dirichlet series. For a particular value of the parameter the series turns into the Dirichlet series for the Riemann zeta function. We prove that almost every series of the family has analytic continuation to the half plane Re s > 1/2 where it doesn't vanish. The result was obtained before by different authors. We give its simple proof in terms of estimates of some trigonometric sums.