  Let K/k be an Abelian extension of number fields, S be a set of places of k,
and p be an odd prime number. We continue an earlier investigation of the
author into the values at zero of the S-imprimitive partial zeta functions of
K/k. An earlier result provides, under the assumption that the p-power roots of
unity in K are cohomologically trivial, a criterion for the values to have
larger than expected p-valuation. The present paper provides such a criterion
for a special class of degree 2p extensions for which the p-power roots of
unity are not cohomologically trivial. For such extensions, new sufficient
conditions are also given for the p-local Brumer-Stark conjecture for K/k and
for Leopoldt's conjecture on the number of independent Zp-extensions of k.
