  We introduce and study the Hilbert space of $(L^2,\Gamma,\chi)$-likewise
theta functions on $\mathbb{R}^d$ with respect to a given discrete subgroup
$\Gamma$ of arbitrary rank and a character $\chi$ of $\Gamma$. A concrete
description is given and an orthonormal basis is then constructed. Its range by
the classical Segal-Bargmann transform is also characterized and leads to the
so-called theta-Bargmann Fock space.
