An Explicit Entire Function of Order One with All Zeros on a Line and Bounded in a Half-Plane

Abstract: We construct a single explicit entire function $Ξ_c(s)$ of order 1, with all zeros provably on $Re(s) = 1/2$, satisfying a functional equation $Ξ_c(s) = Ξ_c(1-s)$, whose normalized form $$Z_c(s) = Ξ_c(s)/[\tfrac{1}{2}s(s-1)π<sup>{-s/2}Γ(s/2)]$$ is uniformly bounded for $Re(s) &gt; 1 + δ$ yet satisfies $\sup_t|Z_c(1+it)| = +\infty$. The function thus satisfies an analogue of the Riemann Hypothesis together with the sharp bounded/unbounded transition at $σ= 1$ characteristic of $ζ$. The construction uses Hadamard products with controlled zero perturbations; the transition is proved unconditionally via a dyadic large-sieve argument and a Phragmén-Lindelöf transfer.