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

We find anЁexplicit formula for anЁarbitrary $6j$-symbol of finite-dimensional irreducible representations of the Lie algebra $\mathfrak{gl}_3$. It is given by the result of substituting $\pm 1$s in a hypergeometric-type series similar to the $\Gamma$-series, which is the simplest several-variate hypergeometric series. We present necessary conditions for the $6j$-symbol to be nonzero.