I think, [here](http://math.stackexchange.com/q/115230/19341), I found 
$$
P_x(s)=\sum_{p < x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n}
\sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}}
\left(
\left[
{\rm li}(t^{\frac zn-s})
\right]^{x}_2
-\left( t^{-s}\operatorname{li}(t^{\frac zn}) \right)_2^x
\right) \tag{7}
$$

where $\rho$ are __all__ the zeros (trivial and non-trivial) of $\zeta$ function.  See the linked question for more detail, corrections are welcome.
Further we know, that
$$
    P(s)=\sum_{n 0}\frac {\mu(n)}n{\log\zeta(ns)} .
$$

So my question is 

> If
$\lim_{x\to \infty} P_x(s)=P(s) $
then
$$
\log\zeta(ns)=\lim_{x\to \infty}
\sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}}
\left(
\left[
{\rm li}(t^{\frac zn-s})
\right]^{x}_2
-\left( t^{-s}\operatorname{li}(t^{\frac zn}) \right)_2^x
\right)
?
$$

**EDIT**

Due to the derivation of $(7)$ (see the [linked question](http://math.stackexchange.com/q/115230/19341)), it doesn't work for $s=0$, but I hope it works for ${\rm Re}(s)=0$.