A Certified Census of the Zeros of the Epstein Zeta Function of m² + 5n² to Height 20,002

A certified census of all zeros of the Epstein zeta function Z(s) = Σ′ (m² + 5n²)−s with 0 < t < 20,002. With K = Q(√−5) and ψ its genus character, Z = ζ(s)L(s,χ−20) + L(s,χ−4)L(s,χ5) = ζK + Lψ, Λ(s) = (√20/2π)sΓ(s)Z(s) = Λ(1−s). Z has no Euler product and has zeros off the critical line (Davenport–Heilbronn 1936). Result for 0 < t < 20,002: 54,524 zeros of Z; 39,870 on the critical line; 7,327 pairs off the line. Every zero is simple. No off-line zero lies within 2−14 of the line (smallest distance 0.001265, at t = 11758.520). Exactly three zeros have real part greater than 1: 1.011049334 + 1193.946405i, 1.015070738 + 4108.108283i and 1.003199405 + 10241.459092i. Running totals: 3,992 zeros below 2,002 and 11,429 below 5,002. No claim is made above 20,002. How it was checked: Arb ball arithmetic (python-flint 0.9.0); 800 windows of length 25 tile [2, 20002]; in each, Backlund counts for Z and its four L-function factors, certified sign changes of the Hardy functions on the line, and winding numbers of Z on cells 1/2 + 2−14 ≤ Re s ≤ 3/2; NZ = SZ + 2NR in every window; each off-line zero certified by a winding number 1 on a square of half-side 2−30. The binary64 Euler–Maclaurin majorant used by the Taylor models is dominated by its Arb bound at all 20,000 cell centres (minimum ratio 1.001), so every certificate stands without a re-run. Thirteen windows needed a recorded repair of the pair locator (the counts were certified before the repair). Limits: computer-assisted; depends on python-flint's zeta and Dirichlet L routines; run in four parts by AI agents on one machine (22–30 October 2026); not re-run by a third party; finite range. Written with Claude (Anthropic), which carried out much of the computation; the human author is responsible for the content. The same verifier design is used in the Davenport–Heilbronn census, doi:10.5281/zenodo.21910254. The supremum of real parts of the zeros of this Z, 1.1339063220928…, is due to Bombieri and Mueller (Forum Math. 20 (2008)).