Lattice Green's Function for the Simple Cubic Lattice in Terms of a Mellin-Barnes Type Integral

The lattice Green's function for the simple cubic lattice I(a)=1π3[triple intergral operator]0πdx dy dza+iε−cos x−cos y−cos zfor a > 3 is expressed as a Mellin-Barnes type integral. The analytic continuation gives simple and useful expressions in series for the numerical calculation of the real part IR(a) and the imaginary part II(a) of the integral for 0 < a < 1 and 1 < a < 3. The values at a = 1, a = 0, and a=√5 are obtained exactly: IR(1) = (π/2)[Γ(5/8)Γ(7/8)]−2, II(1)/(−i)=(√2)IR(1), and II(0) = 3·2−11/3π−4[(Γ(1/3)]6.