  Let $O$ be a one-dimensional Cohen-Macaulay local ring having a finite field
as a coefficient field. The aim of this work is to extend the explicit
computations of the St\"ohr Zeta Function of $O$ for one and two branches to an
arbitrary number of them, obtaining in this general case an upper bound for the
coefficients of the zeta function, instead of an equality. The calculations are
based on the use of the value semigroup of a curve singularity and a suitable
classification of the maximal points of the semigroup.
