The following limit is utilized in Merten’s theorem.
$$ \gamma = \lim\limits_{n\to\infty}\left(-\log\log n - \sum_{p\text{ prime}}^n\log\left(1-\frac{1}{p}\right)\right) $$
I’m interested in the generalization
$$ C_x = \lim\limits_{n\to\infty}\left(x\log\log n - \sum_{p\text{ prime}}^n\log\left(1+\frac{x}{p}\right)\right) $$
For $x\ge 1$. Obviously we can define it in terms of the first equation. In fact for $x=1$ we can say $C_1 = \log\zeta(2) - \gamma$, but for $x\gt 1$ we are left with an unresolved product of primes.
Can we derive a definition of this limit that’s not in terms of primes? Perhaps we can modify a proof of Mertens.
It may be helpful to note that
$$\sum_{k=1}^{\infty} \frac{x^{\Omega(k)}}{k^s} = \prod_{p\text{ prime}}^\infty\left(1-\frac{x}{p^s}\right)^{-1} $$
Adding $x$ times the limit that gives $\gamma$ and the one that gives $C_x$ gives
$$C_x+x\gamma=\lim_{n\to\infty}\left(-\sum_{p\leq n}\log\left(1+\frac xp\right)-x\log\left(1-\frac 1p\right)\right)$$
$$=-\log\left(\prod_p \left(1+\frac xp\right)\left(1-\frac 1p\right)^x\right).$$
This expression is easier to search, and this way I found a few threads mentioning it, notably this answer. It doesn't look like it has a known closed form.