The Lie group \(\mathrm{GL}_3\equiv \mathrm{GL}(3,\mathbb{C})\) and the corresponding Lie algebra \(\mathfrak{gl}_3\equiv\mathfrak{gl}(3,\mathbb{C})\) are important from a mathematical point of view and used in quark theory. The decomposition of the tensor product of representations of \(\mathfrak{gl}_3\) into a direct sum of irreducible ones plays a fundamental role. These decompositions are described by choosing bases in the involved vector spaces and by expending the tensor products of basis vectors. An adequate choice as concerns the realizations of the considered representations and of the used bases leads to simpler expressions for the coefficients of these expressions. These coefficients, called Clebsch-Gordan coefficients, are directly related to the so called \(3j\)-symbols. The author obtains for the \(3j\)-symbols of \(\mathfrak{gl}_3\) simple and explicit formulas in terms of hypergeometric functions by decomposing invariant functions \(f: \mathrm{GL}_3\times \mathrm{GL}_3\times \mathrm{GL}_3\longrightarrow \mathbb{C}\) as linear combinations of products of functions on the factors. The computation is based on an explicit description of the invariant inner product in the space of functions.