## Moments of elliptic integrals and critical $$L$$-values.(English)Zbl 1383.11048

Summary: We compute the critical $$L$$-values of some weight 3, 4, or 5 modular forms, by transforming them into integrals of the complete elliptic integral $$K$$. In doing so, we prove closed-form formulas for some moments of $$K^{\prime 3}$$. Many of our $$L$$-values can be expressed in terms of Gamma functions, and we also obtain new lattice sum evaluations.

### MSC:

 11F03 Modular and automorphic functions 33C20 Generalized hypergeometric series, $${}_pF_q$$ 11M41 Other Dirichlet series and zeta functions 33C75 Elliptic integrals as hypergeometric functions 33E05 Elliptic functions and integrals
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### References:

 [1] Bailey, DH; Borwein, JM, Hand-to-hand combat with multi-thousand-digit integrals, J. Comput. Sci., 3, 77-86, (2012) [2] Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991) · Zbl 0733.11001 [3] Borwein, J.M., Borwein, P.B.: Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley, New York (1987) · Zbl 0611.10001 [4] Borwein, JM; Straub, A; Wan, J, Three-step and four-step random walk integrals, Experiment. Math., 22, 1-14, (2013) · Zbl 1268.33005 [5] Chan, HH; Cooper, S; Liaw, W-C, On $$η ^3(aτ )η ^3(bτ )$$ with $$a+b=8$$, J. Aust. Math. Soc., 84, 301-313, (2008) · Zbl 1151.11018 [6] Duke, W, Some entries in ramanujan’s notebooks, Math. Proc. Camb. Phil. Soc., 144, 255-266, (2008) · Zbl 1230.33009 [7] Glaisher, JWL, On the representation of a number as sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math., 38, 1-62, (1907) [8] Hurwitz, A, Über die entwicklungskoeffizienten der lemniskatischen funktionen, Math. Ann., 51, 196-226, (1899) · JFM 29.0385.02 [9] Kontsevich, M., Zagier, D.: Periods. In: Mathematics unlimited—2001 and beyond, pp. 771-808. Springer, Berlin (2001) · Zbl 1039.11002 [10] Martin, Y, Multiplicative eta quotients, Trans. Amer. Math. Soc., 348, 4825-4856, (1996) · Zbl 0872.11026 [11] Rogers, M.: Hypergeometric formulas for lattice sums and Mahler measures, Intern. Math. Res. Not. 17, 4027-4058 (2011) · Zbl 1282.11099 [12] Rogers, M.: Identities for the Ramanujan zeta function, preprint (2013) [13] Rogers, M., Zudilin, W.: On the Mahler measure of $$1+X+1/X+Y+1/Y$$, Intern. Math. Res. Notices (to appear) · Zbl 1378.11091 [14] Shimura, G, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math., 29, 783-804, (1976) · Zbl 0348.10015 [15] Shimura, G, On the periods of modular forms, Math. Ann., 229, 211-221, (1977) · Zbl 0363.10019 [16] Somos, M.: Dedekind eta function product identities. http://eta.math.georgetown.edu/ · Zbl 0872.11026 [17] Wan, JG, Moments of products of elliptic integrals, Adv. Appl. Math., 48, 121-141, (2012) · Zbl 1231.33020 [18] Zagier, D.: Introduction to modular forms. In: Waldschmidt, M.: et al. (Eds.) From Number Theory to Physics, pp. 238-291. Springer, Heidelberg (1992) · Zbl 0791.11022 [19] Zhou, Y.: Legendre functions, spherical rotations, and multiple elliptic integrals. Ramanujan J. (2013). doi:10.1007/s11139-013-9502-2 [20] Zucker, IJ, Exact results for some lattice sums in 2, 4, 6 and 8 dimensions, J. Phys. A: Math. Nucl. Gen., 7, 1568-1575, (1974) [21] Zucker, IJ, The evaluation in terms of $$Γ$$-functions of the periods of elliptic curves admitting complex multiplication, Math. Proc. Camb. Philos. Soc., 82, 111-118, (1977) · Zbl 0356.33003
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