Glaisher–Kinkelin constant

The Glaisher–Kinkelin constant is defined by the formula $$A=\displaystyle\lim_{n \rightarrow \infty} \dfrac{(2\pi)^{\frac{n}{2}}n^{\frac{n^2}{2}-\frac{1}{12}}e^{-\frac{3n^2}{4}+\frac{1}{12}}}{G(n+1)},$$ where $G$ is the Barnes $G$ function.