Generation of a quasi-contractive semigroup by generalized Ornstein-Uhlenbeck operators ℒ = −∆ + ∇Φ · ∇ − G · ∇ + V + c|x|−2 in the weighted space L2(RN, e−Φ(x)dx) is proven, where Φ ∈ C2(RN, R), G ∈ C1(RN, RN), 0 ≤ V ∈ C1(RN) and c > 0. The proofs are carried out by an application of L2-weighted Hardy inequality and bilinear form techniques.