On the Siegel series in terms of lattice counting

In this paper we describe each coefficient of the Siegel series associated to a quadratic o-lattice L in terms of lattice counting problems, where o is the ring of integers of a non-Archimedean local field of characteristic $0$. Under the restriction that p is odd and that the dimension of the radical of the quadratic space L⊗κ on the residue field κ is at most $2$, we provide explicit values of coefficients and reprove the functional equation of the Siegel series.