On some positivity properties of the Riemann $Ξ$ function

We prove strict log-concavity of the positive functions $Θ$ and $Λ$ with one-sided and bilateral Laplace transforms 1/ξ(12+√ s) and $1/Ξ$, respectively, where Ξ(z)=ξ(12+iz). For either function, log-concavity is the Pólya frequency property of order two (PF₂). Without assuming the Riemann hypothesis (RH), but using its published verification up to height about 3·10¹², we further prove that both densities are PFk_* for k_*=1\,810\,540\,931\,075. The proof views $Θ$ as a small signed perturbation of a density that is PF of every order, with an error bound independent of the order, so the numerical check reduces to scalar inequalities. Since $Λ$ is a Gaussian mixture of $Θ$, it inherits the same order. Conversely, a pole obstruction shows that if either density is PFₖ, then every zeta zero lower than the $(k+2)$-nd lies on the critical line. Finally, we prove that if RH holds up to height T>12, then $Ξ$ satisfies the generalized Laguerre inequalities Lₙ[Ξ](x)≥0 of every order n for |x|≤ T-12, extending the case $n=1$ of Csordas, Ruttan and Varga (1991).