Locating the off-line zeros of the Davenport–Heilbronn function from the primes alone

For the Davenport–Heilbronn function f, every zero ρ with L(ρ,χ)L(ρ,χ̄) ≠ 0 satisfies a phase equation in P = (1/2i) log(L(s,χ)/L(s,χ̄)), and for Re s > 1 that phase is an explicit sum over the primes p ≡ 2, 3 (mod 5) (proved, elementary). Truncating the prime sum to its first K primes, also inside the critical strip where it does not converge, and taking the roots of the truncated equation as predicted zeros uses no zero of any function. In registered tests: at the certified off-line zeros the 300-prime sum matches its target to a median ratio of 0.995, and to 0.996 and 1.000 for two Epstein zeta functions; on the complete certified census of 20,000 ≤ t < 22,000, tested blind, the 300-prime roots recall 77% of the off-line pairs and 98% of those of depth at least 0.2, and shifting the pair list in t drops the recall to zero; at 100,000 ≤ t < 100,500 deep roots refine to certified zeros, and with a winding census of a sub-band this gives 41 certified off-line zeros near t = 105, the deepest 0.9162301512 + 100469.1807065i. We did not find certified strip zeros of this function above t = 22,000 in the literature searched. On the Epstein zeta functions of m² + 5n² and x² + xy + 6y² the same locator finds 99% of the pairs of depth at least 0.2. The locator is a measured fact, not a theorem. Written jointly with Claude, an AI system made by Anthropic, which carried out much of the computation, the derivations and the drafting; the human author is responsible for the content. Not peer reviewed. The package contains the PDF, its LaTeX source, and the scripts, registered predictions and result files it cites.