I've come across some regularized product during study of zeta regularization . We can prove various results like :

$ \infty != \prod_{k=1}^\infty k = \sqrt{2\pi} $

I also know the proof using $\zeta_{\lambda}$ and all the standard stuff . Also , we know ,

$ \infty \# = \prod_{k=1}^\infty p_k = 4\pi^2 $

Where $n\#$ is primorial (product of first $n$ primes)

This is the question asked by C.Soulé, and answered by Garcia and Perez (http://cds.cern.ch/record/630829/files/sis-2003-264.pdf)

Now there are three questions I'd like to ask

(1) is there known intuitive explanation/s for this type of result?

(2) is there known (general )intuitive explanation/s for analytic continuation in general (like we used to compute $\zeta(-1)$ by elementary means ) ?

(3) is it "legal/safe" to say that regularized product of all odd primes is $2π^2$ i.e. can we divide both sides by 2 ?