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Let $L(s)=\sum_{n=1}^{+\infty}\dfrac{a(n)}{n^s}$ be a Dirichlet series were$a(n)$ is a bounded completely multiplicative function. We prove that if $L(s)$extends to a holomorphic function on the open half space $\Re s >1-\delta$,$\delta>0$ and $L(1)=0$ then such a half space is a zero free region of theRiemann zeta function $\zeta(s)$. Similar results is proven for completelymultiplicative functions defined on the space of the ideals of the ring of thealgebraic integers of a number field of finite degree.

If $L(\chi,1)=0$ then $\zeta(1/2+it)\neq0$