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

1,1

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...

Links

Table of n, a(n) for n=1..120.

Wikipedia, Gram points

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.

Example

0.5 + i*3.43621822608696159...

Mathematica

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

Crossrefs

Cf. A058303, A377302.

Sequence in context: A325594 A104076 A238161 * A332880 A281626 A327650

Adjacent sequences: A387390 A387391 A387392 * A387394 A387395 A387396

Keyword

nonn,cons

Author

Jwalin Bhatt, Aug 28 2025

Status

approved