Upper Bounds on the De Bruijn-Newman constant for Dirichlet L-functions with primitive real-valued characters

Let χₘ be a primitive real-valued Dirichlet character of conductor m≥3, let L(s)=L(s,χₘ) be the associated Dirichlet L-function, and let ξL(s) denote its completed form. For each t ∈ R, we define the entire function \[ H^L_t(z) := ∫-∞∞ etu^2\, Φ_L(u)\, eⁱᶻᵘ\, du, \] where ΦL is the superexponentially decaying density \[ Φ_L(u) := 1/2π∫-∞∞ ξ_L\!(1+is/2) e⁻ⁱˢᵘ\, ds. \] The family ₜL ∈ R describes the heat-flow evolution of ξL and solves a backward heat equation. By the work of Newman, there exists a finite constant ΛL, called the generalized de Bruijn--Newman constant, such that all the zeros of HₜL are real if and only if t ≥ ΛL. The Generalized Riemann Hypothesis (GRH) for L is equivalent to the assertion that ΛL ≤ 0. Dobner established the matching lower bound ΛL ≥ 0, while de Bruijn proved the upper bound ΛL ≤ 12. We prove the unconditional uniform bound ΛL≤953/3200=0.2978125 for every primitive real-valued character of conductor 3≤ m≤400, together with stronger individual bounds. The proof extends the effective approximation of Polymath to Dirichlet L-functions, retaining the conductor dependence in the error estimates. We combine these estimates with Platt's numerical verification of GRH by Platt and rigorous interval computations. We also give conditional bounds under explicit global initial zero-strip hypotheses, including |Imz|≤3/4 for the zeros of H₀L.