The $N$-th Hankel relaxation Riemann hypothesis

Let $\rho_k$ be the nontrivial zeros of the Riemann zeta function with positive imaginary part, listed with multiplicity, and define

$$\eta_k=\operatorname{Im}(\rho_k)+i\left(\frac12-\operatorname{Re}(\rho_k)\right),\qquad \lambda_k=\eta_k^2.$$

Let $\Lambda$ be the entire function

$$\Lambda(z)=\prod_k\left(1-\frac{z}{\lambda_k}\right).$$

$N$-th Hankel relaxation Riemann hypothesis. The function $\Lambda$ is in the $N$-th order Laguerre–Pólya class. These are proposed relaxations of the Riemann hypothesis: membership for every $N$ is equivalent to the Riemann hypothesis, while the source does not establish any fixed-$N$ relaxation. This condition is also equivalent to positivity of the associated Hankel matrix $A_N(x)$ for real $x$ away from the zeros.