Let [Formula: see text] be a finite-dimensional representation of the group [Formula: see text] of [Formula: see text] matrices with complex coefficients and determinant one. Let [Formula: see text] be the algebra of [Formula: see text]-invariant polynomials on [Formula: see text]. We present a calculation of the Hilbert series [Formula: see text] as well as formulas for the first four coefficients of the Laurent expansion of [Formula: see text] at [Formula: see text].