  We establish a principal value integral formula, for the residue of the
largest non-trivial candidate pole of the real or complex local zeta function
associated to an analytic germ f, which is non-degenerate with respect to its
Newton polyhedron. In particular, up to an easy non-zero factor, this residue
only depends on the (tau_0)-principal part of f, where tau_0 is the smallest
face of the Newton polyhedron intersecting the diagonal. This formula allows us
to prove some vanishing results for the residue. More precisely, we prove that
the residue vanishes when tau_0 is unstable, and we give a partial proof of the
reverse implication in the complex case. We also deduce an explicit formula for
the residue, in the case where tau_0 is a simplex of codimension 1, and the
only points of the support of f on tau_0 are its vertices.
