  We define an integral version of Sczech's Eisenstein cocycle on GLn by
smoothing at a prime ell. As a result we obtain a new proof of the integrality
of the values at nonpositive integers of the smoothed partial zeta functions
associated to ray class extensions of totally real fields. We also obtain a new
construction of the p-adic L-functions associated to these extensions. Our
cohomological construction allows for a study of the leading term of these
p-adic L-functions at s=0. We apply Spiess's formalism to prove that the order
of vanishing at s=0 is at least equal to the expected one, as conjectured by
Gross. This result was already known from Wiles' proof of the Iwasawa Main
Conjecture.
