Wikipedia has a page about the prime zeta function which is defined as follows:
$P(s)=\sum_{p\;\text{prime}} \frac1{p^s}$
I entered this additional definition:
Define a sequence: $a_n=\prod_{d\mid n} \frac{\Lambda(d)}{\log(d)},$ where zeros are not included in the multiplication and $a_1=1$ then: $P(s)=\log\sum_{n=1}^\infty \frac{a_n}{n^s}.$ Is it a problem that this later definition does not include the zeros in the multiplication?