Bailey, D. H.; Borwein, J. M.; Crandall, R. E.; and Zucker, I. J. "Lattice Sums Arising from the Poisson Equation." J. Phys. A46, 115201, 31 pp., 2013.Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly113, 481-509, 2006.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J. M. and Borwein, P. B. Pi & 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 Constants." J. Phys. Ser. A: Math. and Gen.20, 5497-5510, 1987.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.Havil, J. "Madelung's Constant." §3.4 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 33-35, 2003.Sloane, N. J. A. Sequences A016639, A085469, A086054, and A086055 in "The On-Line Encyclopedia of Integer Sequences."Tyagi, S. "New Series Representation for Madelung Constant." Oct. 17, 2004. http://arxiv.org/abs/cond-mat/0410424.
时求值，被称为马德隆常数。
，可以闭合形式表达的马德隆常数，以下表格中总结了一些示例，其中
是狄利克雷beta函数，
是狄利克雷eta函数。
的闭合形式，将双重求和分解为不包括
的部分，
，因此以上所有项都可以组合成![b_2(2s)=4[sum_(i,j=1)^infty((-1)^(i+j))/((i^2+j^2)^s)+sum_(i=1)^infty((-1)^i)/(i^(2s))].](/images/equations/MadelungConstants/NumberedEquation2.svg)
的情况下，简化为
的情况下，可以写成
由本森公式给出 (Borwein and Bailey 2003, p. 24)
有时被称为“the”马德隆常数，对应于三维 NaCl 晶体的马德隆常数。Crandall (1999) 给出了表达式![M=-2pi+(Gamma(1/8)Gamma(3/8)sqrt(2))/(pi^(3/2))
+2sum^'_(m, n, p=-infty)^infty((-1)^(m+n+p))/(sqrt(m^2+n^2+p^2)[exp(4pisqrt(m^2+n^2+p^2))+1]).](/images/equations/MadelungConstants/NumberedEquation7.svg)
![-1/2-(ln2)/pi-pi/3-2sum^'_(m,n,p=-infty)^infty((-1)^(m+n+p))/(sqrt(m^2+n^2+p^2)[exp(2pisqrt(m^2+n^2+p^2))-1])](/images/equations/MadelungConstants/Inline32.svg)
![sqrt(2)-pi+2sum^'_(m,n,p=-infty)^infty((-1)^(m+n+p))/(sqrt(m^2+n^2+p^2)[exp(2pisqrt(m^2+n^2+p^2))+1])](/images/equations/MadelungConstants/Inline35.svg)
![-1/4-(ln2)/(2pi)-(2pi)/3+1/(sqrt(2))-2sum^'_(m,n,p=-infty)^infty((-1)^(m+n+p))/(sqrt(m^2+n^2+p^2)[exp(4pisqrt(m^2+n^2+p^2))-1])](/images/equations/MadelungConstants/Inline38.svg)
![M=-1/8-(ln2)/(4pi)-(4pi)/3+1/(2sqrt(2))+(Gamma(1/8)Gamma(3/8))/(pi^(3/2)sqrt(2))
-2sum^'_(m,n,p=-infty)^infty((-1)^(m+n+p))/(sqrt(m^2+n^2+p^2)[exp(8pisqrt(m^2+n^2+p^2))-1]),](/images/equations/MadelungConstants/NumberedEquation8.svg)
，没有已知的闭合形式 (Bailey et al. 2006)。
可以闭合形式表达为
