Non vanishing of Dirichlet series of completely multiplicative functions

Let L(s) =sum a(n) be a Dirichlet series where a(n) is a bounded completely multiplicative function. We prove that if L(s) extends to a holomorphic function on the open half space Re s > 1 − , > 0 and L(1) = 0 then such a half space is a zero free region of the Riemann zeta function (s). Similar results are proven for completely multiplicative functions defined on the space of the ideals of the ring of the algebraic integers of a number field of finite degree.