I spent another half hour or so during my lunch break today, and if my extrapolations are even roughly correct, then the point where both products are equal should be about s=1.39773 ± 0.0001.
The easiest way to calculate the series of composites for me is Product (n>=2) {1/(11/n^s)}  Zeta(s), so I don't think I can get any further ATM (maybe another decimal digit or two at most). But it's an interesting problem nonetheless.
