the Madelung constants expressible in closed form for even indices ,
a few examples of which are summarized in the following table, where is the Dirichlet
beta function and is the Dirichlet
eta function.

(OEIS A085469), where the prime indicates that
summation over (0, 0, 0) is excluded.

is sometimes called "the" Madelung constant, corresponds to the Madelung
constant for a three-dimensional NaCl crystal. Crandall (1999) gave the expression

(13)

Similar results were found by Tyagi (2004),

(14)

(15)

(16)

the last of which converges rapidly. Averaging (16) and (13) then gives the beautiful equation

(17)

which is correct to 10 decimal digits even if the sum is completely omitted (Tyagi 2004).

However, no closed form for is known (Bailey et al. 2006).

For hexagonal lattice sums, is expressible in closed form as

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& the AGM: A Study in Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.Buhler, J. and Wagon, S. "Secrets of the
Madelung Constant." Mathematica in Education and Research5, 49-55,
Spring 1996.Crandall, R. E. "New Representations for the Madelung
Constant." Exp. Math.8, 367-379, 1999.Crandall,
R. E. and Buhler, J. P. "Elementary Function Expansions for Madelung
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S. R. Mathematical
Constants. Cambridge, England: Cambridge University Press, 2003.Havil,
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A085469, A086054,
and A086055 in "The On-Line Encyclopedia
of Integer Sequences."Tyagi, S. "New Series Representation
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