I have been able to simplify the product to: $$ \prod _{n=1}^{\infty} \left(2-e^\frac{1}{\left(p_n\right){}^s}\right) = \exp \left(-\sum _{n=1}^{\infty} \frac{\text{Li}_{1-n}\left(\frac{1}{2}\right) P(n s)}{n!}\right)$$ where $p_n$ is the $n^{th}$ prime, $Li_s(x)$ is the Polylogarithm, and $P(s)$ is the Prime Zeta Function. To make sure I hadn't made a mistake the following table gives some values.
\begin{array}{ccc} s & LHS & RHS \\ 3 & 0.823535399 & 0.823535399 \\ 4 & 0.921900096 & 0.921900096 \\ 5 & 0.963887897 & 0.963887897 \\ 6 & 0.982829133 & 0.982829133 \\ \end{array}
The left-hand-side seems so simple so I figured there should at least be a function that makes it pseudo-closed form.