For any periodic function f : N → C with period q, we study the Dirichlet series L(s, f ) := n≥1 f (n)/n s . It is well-known that this admits an analytic continuation to the entire complex plane except at s = 1, where it has a simple pole with residueThus, the function is analytic at s = 1 when ρ = 0 and in this case, we study its non-vanishing using the theory of linear forms in logarithms and Dirichlet L-series. In this way, we give new proofs of an old criterion of Okada for the non-vanishing of L(1, f ) as well as a classical theorem of Baker, Birch and Wirsing. We also give some new necessary and sufficient conditions for the non-vanishing of L(1, f ).