This is hopefully a small point of notation I am missing. I am used to the first two equalities below.
$$\sum_{n \geq 1} \Lambda(n) n^{-s} = \sum_{p \mbox{ prime}} \sum_{m \geq 1} \Lambda(p^m) (p^m)^{-s} = \sum_{p \mbox{ prime}} \sum_{m \geq 1} (\log p) p^{-ms}$$ $$= \sum_{p \mbox{ prime}}(\log p) \sum_{m \geq 1} p^{-ms}. $$
What in the notation of the second sum just above allows us to ascribe values to $p?$ To me it looks like the product:
$$(\log 2+\log3+\log 5+...)(p^{-s}+p^{-2s}+p^{-3s}+...) $$
Apparently we should we read it as (?)
$$\sum_p \left(\log p\sum_{m\geq1}p^{-ms} \right)$$
$$ = (\log 2)(2^{-s}+2^{-2s}+2^{-3s}+...)+(\log 3)(3^{-s}+3^{-2s}+3^{-3s}+...)$$
If so does this second line offer some advantage over the last (double sum) in the first line?
Thanks.