  Let $f(\mathbf z)$ be an analytic function defined in the neighborhood of the
origin of $\mathbb C^n$ which have some Newton degenerate faces. We generalize
the Varchenko formula for the zeta function of the Milnor fibration of a Newton
non-degenerate function $f$ to this case. As an application, we give an example
of a pair of hypersurfaces with the same Newton boundary and the same zeta
function with different tangent cones.
