Li's criterion and zero-free regions of L-functions

Let ζ denote the Riemann zeta function, and let ξ(s)=s(s-1)π-s/2Γ(s/2)ζ(s) denote the completed zeta function. A theorem of X.-J. Li states that the Riemann hypothesis is true if and only if certain inequalities Pn(ξ) in the first n coefficients of the Taylor expansion of ξ at s=1 are satisfied for all n∈N. We extend this result to a general class of functions which includes the completed Artin L-functions which satisfy Artin's conjecture. Now let ξ be any such function. For large N∈N, we show that the inequalities P1(ξ),…,PN(ξ) imply the existence of a certain zero-free region for ξ, and conversely, we prove that a zero-free region for ξ implies a certain number of the Pn(ξ) hold. We show that the inequality P2(ξ) implies the existence of a small zero-free region near 1, and this gives a simple condition in ξ(1), ξ′(1), and ξ″(1), for ξ to have no Siegel zero.