  We obtain the Potential NRQCD Lagrangian relevant for S-wave states with
next-next-to-next-to-leading logarithmic (NNNLL) accuracy. We compute the heavy
quarkonium mass of spin-averaged $l= 0$ (angular momentum) states, with
otherwise arbitrary quantum numbers, with NNNLL accuracy. These results are
complete up to a missing contribution of the two-loop soft running.
