{"schema_version":"dataset-v1","id":"zeta-hardy-z","version":1,"title":"Hardy’s Z function and the first zeta zeros","description":"Values of Hardy’s Z(t) for 0 < t ≤ 100 and the imaginary parts γ of the nontrivial zeros ½ + iγ of ζ(s) in that range. Z(t) is real and |Z(t)| = |ζ(½ + it)|, so each sign change of Z marks a zero on the critical line.","license":"CC0-1.0 (computed values)","problems":["riemann-hypothesis"],"statements":[{"problem":"riemann-hypothesis","statement":"rh-finite-height","role":"All zeros up to height 3,000,175,332,800 are verified on the line."}],"figure":{"family":"function-plot","title":"Zeros on the critical line","caption":"Each sign change of Hardy’s Z(t) is a zero ½ + iγ of ζ on the critical line. This window shows the first zeros; certified verification reaches far higher.","x_label":"t","y_label":"Z(t)","series":"hardy_z","marks":"zeros","mark_label":"γ","window":{"min":20,"max":100,"step":5,"default":100}},"provenance":{"method":"mpmath siegelz(t) on a 0.1 grid and zetazero(n) for every zero with imaginary part at most 100, then rounded for display.","software":{"mpmath":"1.3.0","python":"3.12.13"},"precision":"25 significant digits; Z rounded to 5 and γ to 9 decimal 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