An explicit formula for the local zeta function of a Laurent polynomial

In a recent paper Zúñiga-Galindo and the author begun the study of the local zeta functions for Laurent polynomials. In this work we continue this study by giving a very explicit formula for the local zeta function associated to a Laurent polynomial f over a p-adic field, when f is weakly non-degenerate with respect to the Newton polytope of f at infinity.