It is well-known that the discrete version of Stirling's formula is as follows
$$ \log N!=\left(N+\frac12\right)\log N-N+\log C+\int_N^\infty{P_1(t)\over t}\mathrm dt\tag1 $$
where $P_1(t)=B_1(\{t\})$, and $C$ is some constant term emerged while applying the Euler-Maclaurin formula. Using the limit definition of gamma function that
$$ \Gamma(s+1)=\lim_{N\to\infty}{N^sN!\over(s+1)(s+2)\cdots(s+N)} $$
I am able to obtain an identity similar to (1) but with respect to continuous variable $s$:
$$ \log\Gamma(s+1)=\left(s+\frac12\right)\log s-s+\log C+\int_0^\infty{P_1(t)\over s+t}\mathrm dt\tag2 $$
I already know that $C=\sqrt{2\pi}$ can be obtained by replacing factorial terms in Wallis product, but I wonder if there is an alternative way to figure out the constant.