  We introduce the zeta function of the prehomogenous vector space of binary
cubic forms, twisted by the real analytic Eisenstein series. We prove the
meromorphic continuation of this zeta function and identify its poles and their
residues. We also identify the poles and residues of the zeta function when
restricted to irreducible binary cubic forms. This zeta function can be used to
prove the equidistribution of the lattice shape of cubic rings.
