The Hilbert series of ${\mathrm{SL}}_2$-invariants

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