Exact certificates of linear independence for the low zeros of ζ(s) and Dirichlet L-functions

We give exact, fully verified certificates that no integer linear relation with bounded coefficients holds among the low nontrivial zeros of the Riemann zeta function, and—apparently for the first time—among the zeros of several Dirichlet L-functions jointly. For the ordinates γ1, . . . , γ16 of ζ(s) we certify the absence of relations with coefficients up to 1.28 × 1048; analogous certificates are given for the first 32, 48, 64, 96 and 128 ordinates and for sliding windows through γ128. Across L-functions we certify joint independence fragments for ζ(s), L(s, χ−4), L(s, χ−3) and L(s, χ5): all six pairwise certificates at heights ≈ 5×1014 and a joint certificate over all 48 low zeros at height 1,338,277. These are finite fragments of the Linear Independence conjecture and of the Grand Simplicity Hypothesis of Rubinstein–Sarnak; within the certified envelope, classical conditional results of Grosswald and the prime-race framework rest on verified hypotheses. The certificate chain contains no floating-point step: after an LLL reduction, Gram–Schmidt norms are computed as exact ratios of integer minor determinants, and the certified height is obtained by exact integer search. The only assumption is the accuracy of the computed zeros, which is supported by two independent implementations and, for the L-zeros, by two independent formulas.