  Let $V$ be a finite dimensional representations of the group
$\operatorname{SL}_2$ of $2\times 2$ matrices with complex coefficients and
determinant one. Let $R=\mathbb{C}[V]^{\operatorname{SL}_2}$ be the algebra of
$\operatorname{SL}_2$-invariant polynomials on $V$. We present a calculation of
the Hilbert series $\operatorname{Hilb}_R(t)=\sum_{n\ge 0}\dim (R_n)\: t^n$ as
well as formulas for the first four coefficients of the Laurent expansion of
$\operatorname{Hilb}_R(t)$ at $t=1$.
