On the Siegel series in terms of lattice counting

In this paper we describe each coefficient of the Siegel series associated to a quadratic $\mathfrak{o}$-lattice $L$ in terms of lattice counting problems, where $\mathfrak{o}$ 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 $L\otimes\kappa$ on the residue field $\kappa$ is at most $2$, we provide explicit values of coefficients and reprove the functional equation of the Siegel series.