Riemann Zeta

Real Riemann zeta

For s>1, ζ(s) is the sum of n^(−s). Analytic continuation defines the other real values, except the single pole at s=1. Negative even integers −2,−4,… are exact trivial zeros. This real-line tool does not show nontrivial complex zeros.

Choose an overview, negative-zero, positive-value or pole view. Click or drag to evaluate a point; red zeros are clickable. The numeric field can evaluate outside the plotted interval. Plot bounds are limited to[−300,1000]. Only the true pole and numerical overflows break the sampled curve; no artificial near-pole band is removed.

The signed logarithmic axis uses sign(ζ)·log10(1+|ζ|), with ticks labeled in ζ values. It retains negative values and zero while showing large magnitudes. The linear axis is also available; very large linear values use an explicitly labeled common scale factor.

Calculations use double precision, Euler–Maclaurin with a bounded correction series, and the functional equation in the log domain. Finite neighbors of the pole and zeros are not snapped to them. Overflow differs from a pole. Values at sufficiently large positive s round to1 at floating-point precision. The sampled plot and displayed digits are approximations, not certified error bounds.

DLMF 25.2 · DLMF 25.4

Views and scale choices update directly; range edits follow after a short pause or Enter. Click or drag the plot to choose s, or select an exact negative even root from the list. The compact ±log legend denotes sign(y) log10(1+|y|); tick labels still show original ζ values.

A completed session reconstructs the bounded real-axis samples with the same numerical method; it does not restore a complex-plane plot or certify every point between samples. Unfinished work stays unfinished. Numeric drafts are limited to128 characters.

Only input values are remembered in this browser. Calculated results and experiment histories are not saved.