An equation derived by Kronecker: sum'_(x, y, z = - ∞)^∞ (x^2 + y^2 + d z^2)^(-s) = 4ζ(s) η(s) + (2π)/(s - 1) (ζ(2s - 2))/d^(s - 1) + (2π^s)/(Γ(s)) d^((1 - s)/2) sum_(n = 1)^∞ n^((s - 1)/2) sum_(u^2|n) (r(n/u^2))/u^(2s - 2) integral_0^∞ e^(π sqrt(n d)(y + y^(-1))) y^(s - 2) d y, where r(n) is the sum of squares function, ζ(z) is the Riemann zeta function, η(z) is the Dirichlet eta function, Γ(z) is the gamma function, and the primed sum omits terms with zero denominator.

Dirichlet eta function | Epstein zeta function | lattice sum | sum of squares function