Let ζ(s) = σn-s (Res >1) denote the Riemann zeta function; then, as is well known, , where Bm denotes the mth Bernoulli number, In this paper we investigate the possibility of similar evaluations of the Epstein zeta function ζq(s) at the rational integers s = k> 2. Let
be a positive definite quadratic form and
where the summation is over all pairs of integers except (0, 0). In attempting to evaluate ζq(k) we are guided by Kronecker's first limit formula 
where γ is Euler's constant,
is the Dedekind eta-function, and τ is the complex number in the upper half plane, ℋ, associated with Q by the formula
On the basis of (1.3) we would expect a formula involving functions of τ. This formula is stated in Theorem 1, (2.13).