  We study the class numbers of integral binary cubic forms. For each $SL_2(Z)$
invariant lattice $L$, Shintani introduced Dirichlet series whose coefficients
are the class numbers of binary cubic forms in $L$. We classify the invariant
lattices, and investigate explicit relationships between Dirichlet series
associated with those lattices. We also study the analytic properties of the
Dirichlet series, and rewrite the functional equation in a self dual form using
the explicit relationship.
