A recursion equation for prime numbers

It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$p_{n+1} =\lim\limits_{s\to +\infty} [\prod\limits^n_{k=1} (1-\frac{1}{p^s_k}) \sum\limits^\infty_{j=1} \frac{1}{j^s} -1]^{-1/s}.$$ The upper limit on the sum can be replaced by $2p_n -1$, and the result still holds.