What would be the asymptotic growth rate of $\displaystyle \zeta _{p}{(s)}$ or for the special cases $\displaystyle \zeta _{p}{(2)}$ or $\displaystyle \zeta _{p}{(3)}$

I tried using the relation $\displaystyle {p}_{n}\sim n\ln (n)$ but couldn't get it to work. How many terms would you require to compute $\displaystyle \zeta _{p}{(2)}$ and $\displaystyle \zeta _{p}{(3)}$ to 20 digits using direct computation??