The authors study the Hankel determinant \(D_{n}(t)\) of the generalized Jacobi weight \(w(x):=w(x;t)=(x-t)^{\gamma} x^{\alpha} (1-x)^{\beta}\), \(x\in[0,1]\) with \(\alpha,\beta>0\), \(t<0\) and \(\gamma \in \mathbb R\). The motivation for this research mainly arises from the close relation between Hankel determinants and random matrix theory, which is of interest in mathematical physics. The authors investigate the properties of \(D_{n}(t)\) as a function of \(t\). More precisely, based on the ladder operators for the corresponding monic orthogonal polynomials \(P_{n}(x)\), they show that the logarithmic derivative of the Hankel determinant \(D_{n}(t)\) is characterized by a Jimbo-Miwa-Okamoto \(\sigma\)-form of the Painlevé VI system.