A251809 Decimal expansion of 3*sqrt(2)*Pi^3/128.

1, 0, 2, 7, 7, 2, 2, 5, 8, 5, 9, 3, 6, 8, 5, 8, 5, 6, 7, 8, 7, 9, 2, 5, 6, 6, 1, 8, 0, 0, 2, 2, 5, 5, 7, 6, 7, 2, 1, 0, 1, 0, 0, 3, 1, 8, 5, 3, 6, 9, 9, 7, 4, 6, 5, 3, 3, 1, 0, 8, 4, 7, 5, 5, 1, 8, 5, 2, 5, 7, 7, 7, 2, 4, 6, 8, 5, 8, 4, 9, 6, 8, 0, 3, 5, 1

Offset: 1

Author

Bruno Berselli, Dec 10 2014

Keywords

• nonn
• cons

Comments

Equals the value of the Dirichlet L-series of the non-principal character modulo 8 (A188510) at s=3. - Jianing Song, Nov 16 2019

Examples

			1.027722585936858567879256618002255767210100318536997465331084755185...
		

References

• L. B. W. Jolley, Summation of series, Dover Publications Inc. (New York), 1961, p. 64 (formula 340).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, Section 2.2 L(m=8, r=4, s=3).
• Index entries for transcendental numbers.

Crossrefs

Cf. A153071: Sum_{i >= 0} (-1)^i/(2i+1)^3.

Cf. A233091: Sum_{i >= 0} 1/(2i+1)^3.

Cf. A093954, A188510.

Programs

Magma

R:= RealField(); 3*Sqrt(2)*Pi(R)^3/128; // G. C. Greubel, Jul 27 2018

Mathematica

RealDigits[3 Sqrt[2] Pi^3/128, 10, 90][[1]]

PARI

3*sqrt(2)*Pi^3/128 \\ G. C. Greubel, Jul 27 2018

Formula

Equals Sum_{i >= 0} (-1)^floor(i/2)/(2i+1)^3 = +1 +1/3^3 -1/5^3 -1/7^3 +1/9^3 +1/11^3 - ...

Equals Sum_{i >= 1} A188510(i)/i^3 = Sum_{i >= 1} Kronecker(-8,i)/i^3. - Jianing Song, Nov 16 2019

Equals 1/(Product_{p prime == 1 or 3 (mod 8)} (1 - 1/p^3) * Product_{p prime == 5 or 7 (mod 8)} (1 + 1/p^3)). - Amiram Eldar, Dec 17 2023