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on de Bruijn–Newman constant ($C_{21}$)

The statement

Define

$H(\lambda, z):=\int_{0}^{\infty} e^{\lambda u^{2}} \Phi(u) \cos (z u) \, du$ ,

where $\Phi$ is the super-exponential decaying function

$\Phi(u) = \sum_{n=1}^{\infty} (2\pi^2n^4e^{9u}-3\pi n^2 e^{5u} ) e^{-\pi n^2 e^{4u}}.$

Newman showed in [N1976] that there exists a finite constant $C_{21}$ (the de Bruijn–Newman constant) such that the zeros of $H$ are all real precisely when $\lambda \geq C_{21}$ .

What gets accepted

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Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.