From Legendre's Theorem we know
$$n!=\prod_{p}p^{\lceil \frac{n}{p}\rceil +\lceil \frac{n}{p^{2}}\rceil +.. }$$.
Since $\Gamma (n+1)=n!$, i wonder if there is a generalization of this formula for gamma function. Something like when $x$ is a real number,
$$\Gamma (x+1)=\prod_{p}p^{f_{p}(x) }$$
where i don't know what $f_{p}(x)$ could be.
On
Possible but not very interesting without more conditions. Totally trivial example: $$\Gamma(x+1) = 2^{\log_2\Gamma(x+1)}\,3^0\,5^0\cdots$$ Another almost trivial example: rewrite the product $$\Gamma(x) = \frac{e^{-\gamma x}}x\prod_{n=1}^\infty(1+x/n)^{-1}e^{x/n}$$ as $$\prod_{n=1}^\infty p_n^{\text{something}_{p_n}(x)}$$
a reason why $$n!=\prod_{p}p^{\lfloor \frac{n}{p}\rfloor +\lfloor \frac{n}{p^{2}}\rfloor +.. } $$ is that $$\ln( \lfloor x \rfloor ! ) = \sum_{k \le x} \ln k$$ thus
$$\zeta'(s) = -\sum_{n=1}^\infty n^{-s} \ln n = -s\int_1^\infty \left( \sum_{k \le x} \ln k \right) x^{-s-1}dx = -s\int_1^\infty \ln( \lfloor x \rfloor ! )x^{-s-1}dx = \frac{-\zeta'(s)}{\zeta(s)} \zeta(s)$$
which by inverse Mellin transform gives the desired relation :
$$\ln n! = \sum_{k=1}^n \lfloor n / k \rfloor \Lambda(k) = \sum_{p^k \le n} \lfloor n / p^k \rfloor \ln p$$
where $\Lambda(k)$ is the von Mangoldt function which appears in $-\frac{\zeta'(s)}{\zeta(s)} = \sum_n \Lambda(n) n^{-s}$.
but you can see that all this involves only $\ln( \lfloor x \rfloor ! ) = \ln \Gamma(\lfloor x \rfloor +1)$.
if you want a formula involving $\ln \Gamma(\lfloor x \rfloor +1+\epsilon) $ you'll have in fact to repeat the same process but replacing $\zeta(s)$ with the Hurwitz zeta function $\zeta(s,\epsilon) = \sum_n (n+\epsilon)^{-s}$ and it won't be related to prime numbers anymore because :
$$\text{the Hurwitz zeta function has no Euler product.}$$