Simple question (but not for me): Does the Euler product formula diverge for any zero of the Riemann zeta function?

The reason why I ask this is that I heard we should not use the Euler product instead of the Riemann zeta function for Re(s)=<1 because it diverges on the critical strip, but I am not sure of that.

According to my numerical calculation, it seems that it converges for the (known) zeta zeros.

Additional Question 1: Is it clear that the Euler product for a nontrivial zeta zero is either divergent or convergent?

Additional Question 2: If only one case is possible, which one is the right answer? Divergent or convergent?