A365317 -id:A365317 - cp's OEIS frontend

A365318 Decimal expansion of negative imaginary part of Gamma(exp(i*Pi/3)).

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

Offset: 0

Artur Jasinski, Sep 01 2023

			0.51727909947484...
Gamma(cos(Pi/3) + I*sin(Pi/3)) = 0.37980489179139...-I*0.51727909947484...

Juan Arias de Reyna and Jan van de Lune, 2013, On the exact location of the non-trivial zeros of Riemann's zeta function, arXiv:1305.3844 [math.NT], 2013, formula (4).
Eric Weisstein's World of Mathematics, Riemann-Siegel Functions.
Wikipedia, Riemann-Siegel theta function.

Cf. A212877, A212878, A212879, A212880.

Cf. A365317 (real part), A365319 (abs).

RealDigits[-Im[Gamma[Cos[Pi/3] + I Sin[Pi/3]]], 10, 106][[1]]
(* or *)
RealDigits[Sqrt[Pi/Cosh[Pi Sqrt[3]/2]] Sin[2 RiemannSiegelTheta[Sqrt[3]/2] + ArcTan[Tanh[Pi Sqrt[3]/4]] + Sqrt[3] Log[2 Pi]/2], 10, 106][[1]]

-imag(gamma(exp(I*Pi/3))) \\ Michel Marcus, Sep 01 2023

Equals sqrt(Pi*sech(Pi*sqrt(3)/2))*sin(2*theta(sqrt(3)/2)+(sqrt(3)/2)*log(2*Pi)+arctan(tanh(Pi*sqrt(3)/4))) where theta is Riemann-Siegel theta function.

A364821 Decimal expansion of the unique value of x such that Gamma(x + i*sqrt(1-x^2)) is an imaginary number and -1 < x < 1.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Oct 07 2023

Gamma(A364821 + i*sqrt(1-A364821^2)) = -i*0.5377003887835295919... see A366345.

Also decimal expansion of the unique value of x in the range -1 < x < 1 for which the function Im(Gamma(x + i*sqrt(1-x^2)))/abs(Gamma(x + i*sqrt(1-x^2))) is minimized.

			0.149965974606491...

Cf. A090986, A212877, A212878, A212880, A212879, A364355, A364356, A365317, A365318, A366345.

RealDigits[
  x /. FindRoot[Re[Gamma[x + I Sqrt[1 - x^2]]], {x, 0.15},
    WorkingPrecision -> 106]][[1]]

Last digit corrected by Vaclav Kotesovec, Sep 02 2025

A365319 Decimal expansion of abs(Gamma(exp(i*Pi/3))).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Sep 01 2023

Also abs(Gamma(exp(i*2*Pi/3))).
For real part of Gamma(exp(i*Pi/3)) see A365317.
For negative imaginary part of Gamma(exp(i*Pi/3)) see A365318.

			0.641739372784755...

Juan Arias de Reyna and Jan van de Lune, On the exact location of the non-trivial zeros of Riemann's zeta function, arXiv:1305.3844 [math.NT], 2013, formula (4).

Cf. A109219, A212877, A212878, A212879, A212880, A365317, A365318.

RealDigits[Abs[Gamma[Cos[Pi/3] + I Sin[Pi/3]]], 10, 106][[1]]
(* or *)
RealDigits[Sqrt[Pi/Cosh[Pi Sqrt[3]/2]], 10, 106][[1]]

abs(gamma(exp(I*Pi/3))) \\ Michel Marcus, Sep 01 2023

Equals sqrt(Pi/cosh(Pi*sqrt(3)/2)).

Equals 1/sqrt(3*A109219).

A366345 Decimal expansion of y such that Gamma(A364821 + isqrt(1-A364821^2)) = -yi.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Oct 07 2023

x = A364821 = 0.149965974606491089853... is the only real number x such that Gamma(x + i*sqrt(1-x^2)) is an imaginary number and -1 < x < 1.

			Gamma(A364821 + i*sqrt(1-A364821^2)) = -i*0.5377003887835295919046...

Cf. A090986, A212877, A212878, A212880, A212879, A364355, A364356, A365317, A365318, A364821.

A366545 Decimal expansion of the value x for which the function Re(Gamma(-x + i*sqrt(1-x^2))) is minimized for -1 < x < 1.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Oct 12 2023

For Re(Gamma(-A366545 + i*sqrt(1-A366545^2))) = -0.930840199... see A366545.

			0.9565130903466545656...

Cf. A090986, A212877, A212878, A212880, A212879, A364355, A364356, A364821, A365317, A365318, A366345.

xmin = Re[x /. FindRoot[1/(2 Sqrt[1 - x^2]) I (Gamma[1 + x - I Sqrt[1 - x^2]] PolyGamma[0, x - I Sqrt[1 - x^2]] - Gamma[1 + x + I Sqrt[1 - x^2]] PolyGamma[0,
         x + I Sqrt[1 - x^2]]), {x, -0.98}, WorkingPrecision -> 110]];
 RealDigits[xmin, 10, 106][[1]]

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A365318 Decimal expansion of negative imaginary part of Gamma(exp(i*Pi/3)).

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A364821 Decimal expansion of the unique value of x such that Gamma(x + i*sqrt(1-x^2)) is an imaginary number and -1 < x < 1.

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A365319 Decimal expansion of abs(Gamma(exp(i*Pi/3))).

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A366345 Decimal expansion of y such that Gamma(A364821 + i*sqrt(1-A364821^2)) = -y*i.

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A366545 Decimal expansion of the value x for which the function Re(Gamma(-x + i*sqrt(1-x^2))) is minimized for -1 < x < 1.

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A366345 Decimal expansion of y such that Gamma(A364821 + isqrt(1-A364821^2)) = -yi.