When is an infinite product of natural numbers regularizable?

Given an increasing sequence $0<\lambda_1<\lambda_2<\lambda_3<\ldots$ one defines the regularized infinite product $$ \prod^{\infty}_{n=1}\lambda_n=\exp\left(-\zeta'_{\lambda}(0)\right), $$ where $\zeta_{\lambda}$ is the zeta function associated to the sequence $(\lambda_n)$, $$ \zeta_{\lambda}(s)=\sum^{\infty}_{n=1}\lambda_n^{-s}. $$ (See the paper: E.Munoz Garcia and R.Perez-Marco."The Product over all Primes is $4\pi^2$". http://cds.cern.ch/record/630829/files/sis-2003-264.pdf )

In the paper( https://arxiv.org/ftp/arxiv/papers/0903/0903.4883.pdf ) I have evaluated the $$ \prod_{p-primes}p^{\log p}=\prod_{p-primes}e^{\log^2p}=\exp\left(24\zeta''(0)+12\log^2(2\pi)\right) $$ where $\zeta''(0)$ is the second derivative of Riemann's Zeta function in $0$.