A356092 Decimal expansion of the imaginary part of the first nontrivial zero of zeta'.

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

Offset: 2

Author

Benoit Cloitre, Aug 13 2022

Keywords

• nonn
• cons

Comments

The nontrivial zero of zeta' with the smallest imaginary part is 2.4631618694543212... + i*23.2983204927628579...

The Riemann Hypothesis is equivalent to the assertion that zeta' has no nontrivial zero in the half-plane Re(z) < 1/2 (there are trivial zeros, e.g., -2.717262829204574...).

Links

• Benoit Cloitre, Picture of some nontrivial zeros of zeta'.
• Norman Levinson and Hugh L. Montgomery, Zeros of the derivatives of the Riemann zeta-function, Acta Mathematica, Vol. 133 (1974), pp. 49-65; alternative link.

Crossrefs

Cf. A356216.

Programs

Mathematica

RealDigits[Im[x /. FindRoot[Derivative[1][Zeta][x], {x, 2 + 23*I}, WorkingPrecision -> 100]]][[1]] (* Amiram Eldar, Aug 14 2022 *)