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 [11]
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).