A Painlevé equation for the Hörmander-Bernhardsson constant

The Hörmander-Bernhardsson constant $\mathscr{C}$ is the sharp constant in $|f(0)|\le \mathscr{C} \|f\|_1$ for entire functions of exponential type $\le π$. We prove that $\mathscr{C} = 2πθ_*^{-2}$ where $θ_*$ is the least positive singularity of the regular solution $v$ with $v(0)=0$ of the cosh-Go