  We show, using the quench action approach [Caux&Essler Phys. Rev. Lett. 110
(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.
