Summary: We show, using the quench action approach [J.-S. Caux and F. H. L. Essler, “Time evolution of local observable after quenching to an integrable model”, Phys. Rev. Lett. 110, 257203 (2013)], that the whole post-quench time evolution of an integrable system in the thermodynamic limit can be computed with a minimal set of data which are encoded in what we denote the generalized single-particle overlap coefficient \({s}_{0}^{\Psi_0}(\lambda)\). This function can be extracted from the thermodynamically leading part of the overlaps between the eigenstates of the model and the initial state. For a generic global quench the shape of \({s}_{0}^{\Psi_{0}}(\lambda)\) in the low momentum limit directly gives the exponent for the power law decay to the effective steady state. As an example we compute the time evolution of the static density-density correlation in the interacting Lieb-Liniger gas after a quench from a Bose-Einstein condensate. This shows an approach to equilibrium with power law \(t^{-3}\) which turns out to be independent of the post-quench interaction and of the considered observable.