We compute the 0-, 1-, and
2-dimensional homology of the vector field Lie algebra L2 with coefficients in the
modules ℱλ,μ, and conjecture that the higher dimensional homology for any λ and μ
is zero. We completely compute the 0- and 1-dimensional homology with coefficients
in the more complicated modules Fλ,μ. We also give a conjecture on this homology in
any dimension for generic λ and μ.