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

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

Author

Jwalin Bhatt, Aug 28 2025

Keywords

• nonn
• cons
• new

Comments

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...

Examples

			0.5 + i*3.43621822608696159...
		

Links

• Wikipedia, Gram points

Crossrefs

Cf. A058303, A377302.

Programs

Mathematica

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

Formula

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.