Cubic lattice sums include the following:(1)(2)(3)where the prime indicates that the origin , , etc. is excluded from the sum (Borwein and Borwein 1986, p. 288).These have closed forms for even ,(4)(5)(6)(7)for , where is the Dirichlet beta function, is the Dirichlet eta function, and is the Riemann zeta function (Zucker 1974, Borwein and Borwein 1987, pp. 288-301). The lattice sums evaluated at are called the Madelung constants. An additional form for is given by(8)for , where is the sum of squares function, i.e., the number of representations of by two squares (Borwein and Borwein 1986, p. 291). Borwein and Borwein (1986) prove that converges (the closed form for above does not apply for ), but its value has not been computed. A number of other related double series can be evaluated analytically.For hexagonal sums, Borwein and Borwein (1987, p. 292) give(9)where . This Madelung constant is expressible in closed..