Effective Estimates for Generalized Merten's Theorem

Question

Fix $r>1$. I would like to obtain strong effective estimates for $$\prod_{p\leq x}\left(1-p^{-r}\right).$$ Certainly this product tends to $1/\zeta(r)$ as $x\to\infty$, but I am more interested in how quickly it does so. I assume I could adapt the proof of Merten's Third theorem (which is the case $r=1$) in order to get some reasonable asymptotic estimate. However, since I am more interested in effective estimates (and I don't want to spend an enormous amount of time trying to optimize constants), I was hoping someone might know of a reference for this.