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.
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