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ZetaZero

ZetaZero[k]
represents the k

zero of the Riemann zeta function on the critical line.

ZetaZero[k, t]
represents the k

zero with imaginary part greater than

For positive k, ZetaZero[k] represents the zero of on the critical line that has the k smallest positive imaginary part.

For negative k, ZetaZero[k] represents zeros with progressively larger negative imaginary parts.

Find numerically the position of the first zero:

Symbolic property:

Find numerically the position of the first zero:

Symbolic property:

Evaluate to high precision:

Find the first zero with the imaginary part greater than 1000:

ZetaZero threads element-wise over lists:

Negative order is interpreted as a reflected root of the Zeta function:

Plot distances between successive zeros:

Visualize the first ten zeros:

Show good Gram points, where RiemannSiegelZ changes sign for consecutive points:

Show a bad Gram point:

First occurrence of Lehmer's phenomenon:

ZetaZero[0] is not defined: