Classical $6j$-symbols for finite dimensional representation of the algebra $\mathfrak{gl}_3$

In the paper an explicit formula for an arbitrary $6j$-symbol for finite-dimensional irreducible representations of the algebra $\mathfrak{gl}_3$ is derived. A $6j$-symbol is written as a result of substitution of $\pm 1$ into a series of hypergeometric type, which is similar to a $\Gamma$-series, which is a simplest example of a multivariate series of hypergeometric type. The selection rulers for a $6j$-symbol are derived.