The aim of this paper is to introduce and investigate the Poincar\'e series
associated with the Weierstra{\ss} semigroup of one and two rational points at
a (not necessarily irreducible) non-singular projective algebraic curve defined
over a finite field, as well as to describe their functional equations in the
case of an affine complete intersection.Comment: Beginning of Section 3 and Subsection 3.1 were modifie