A Li-type criterion for zero-free half-planes of Riemann's zeta function

We define a sequence of real functions which coincide with Li’s coefficients at one and which allow us to extend Li’s criterion for the Riemann Hypothesis to yield a necessary and sufficient condition for the existence of zero-free strips inside the critical strip $0<\mathrm{\Re }(z)<1$. We study some of the properties of these functions, including their oscillatory behaviour.