A377302 -id:A377302 - cp's OEIS frontend

A377297 Decimal expansion of the smallest positive imaginary solution to Gamma(1+z) = Gamma(1-z).

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

Offset: 1

Jwalin Bhatt, Oct 23 2024

All solutions are either purely real or purely imaginary. The smallest solution (by absolute value) happens to be purely imaginary.
When expressed in terms of Gauss's Pi function, it is
- The smallest solution to Pi(z) = Pi(-z).
- The smallest `y` such that: Pi(i*y) is purely real or, equivalently, Gamma(i*y) is purely imaginary.
- Arg(Pi(i*y)) is given by Sum_{n>=1} y/n - arctan(y/n) - y*euler_gamma, so for Pi(i*y) to be purely real the argument must equal 2*Pi*k where k in some integer. - Jwalin Bhatt, Aug 23 2025

			1.8055470716051069198763666... .

Cf. A000796, A001620, A212880, A377302.

Digits:= 120:
fsolve(GAMMA(1+z*I)=GAMMA(1-z*I), z=0..3);  # Alois P. Heinz, Oct 25 2024

RealDigits[x /. FindRoot[Gamma[1 + x*I] == Gamma[1 - x*I], {x, 2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 23 2024 *)
RealDigits[x /. FindRoot[Re[Gamma[I*x]] == 0, {x, 2}, WorkingPrecision -> 120]][[1]] (* Vaclav Kotesovec, Oct 25 2024 *)

from mpmath import mp, nstr, factorial, findroot
mp.dps = 120
root = findroot(lambda z: factorial(z)-factorial(-z), 1.8j).imag
A377297 = [int(d) for d in nstr(root, n=mp.dps)[:-1] if d != '.']

Gamma(1+i*1.8055470716051069...) = Gamma(1-i*1.8055470716051069...) = 0.19754864094576264...
From Jwalin Bhatt, Aug 23 2025: (Start)
Smallest positive imaginary root of the equation x*sin(Pi*x)*Gamma(x)^2 = Pi.
Smallest positive real root of the equation Sum_{n>=1} x/n - arctan(x/n) = x * gamma where gamma = A001620. (End)

A387378 Decimal expansion of the smallest positive real solution > 0.5 to zeta(z) = zeta(1-z).

Original entry on oeis.org

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

Offset: 2

Jwalin Bhatt, Aug 28 2025

Using the reflection formula for the zeta function, one can also rewrite the equality in terms of the Gamma function as Gamma(z) = (2^(z-1))*(Pi^z)*sec((Pi*z)/2).
There are infinitely many solutions on the real axis and on the critical line.
The solutions on the critical line are the gram points.
There are 12 complex solutions apart from these out of which 3 are unique:
   8.990914533614919... + i*4.510594140699146...
  13.162787864991035... + i*2.580464971850669...
  16.478090665944547... + i*0.679406009477847...

			19.06775084706966207279...

Cf. A058303, A365281, A377302.

RealDigits[x /. FindRoot[Zeta[x] == Zeta[1 - x], {x, 19}, WorkingPrecision -> 120]][[1]]

zeta(19.067750847069662...) = zeta(1-19.067750847069662...) = 1.000001820649741...
Smallest positive real root > 0.5 of the equation Gamma(z) = (2^(z-1))*(Pi^z)*sec((Pi*z)/2).
Equals A365281 + 1/2. - Amiram Eldar, Aug 28 2025

A387393 Decimal expansion of the imaginary part of the smallest complex solution to zeta(z) = zeta(1-z).

Original entry on oeis.org

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

Offset: 1

Jwalin Bhatt, Aug 28 2025

Using the reflection formula for the zeta function, one can also rewrite the equality in terms of the Gamma function as Gamma(z) = (2^(z-1))*(Pi^z)*sec((Pi*z)/2).
There are infinitely many solutions on the real axis and on the critical line.
The solutions on the critical line are the gram points and this is the first positive gram point.
There are 12 complex solutions apart from these out of which 3 are unique:
   8.990914533614919... + i*4.510594140699146...
  13.162787864991035... + i*2.580464971850669...
  16.478090665944547... + i*0.679406009477847...

			0.5 + i*3.43621822608696159...

Wikipedia, Gram points

Cf. A058303, A377302.

RealDigits[Im[x /. FindRoot[Zeta[x] == Zeta[1 - x], {x, 0.5+3.5I}, WorkingPrecision -> 20]]][[1]]

zeta(0.5+i*3.436218226086961...) = zeta(0.5-i*3.436218226086961...) = 0.564150979455795...
Smallest complex root > 0.5 of the equation Gamma(z) = (2^(z-1))*(Pi^z)*sec((Pi*z)/2).
Smallest positive zero of sin(theta(t)) where theta is Riemann-Siegel theta function.
Smallest positive root of (-0.5i)*(loggamma(.25+(i*z)*.5)-loggamma(.25-(i*z)*.5)) - (z*log(Pi))*.5 = -Pi.

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A377297 Decimal expansion of the smallest positive imaginary solution to Gamma(1+z) = Gamma(1-z).

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A387378 Decimal expansion of the smallest positive real solution > 0.5 to zeta(z) = zeta(1-z).

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A387393 Decimal expansion of the imaginary part of the smallest complex solution to zeta(z) = zeta(1-z).

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Mathematica

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