We consider the generalized Jacobi weight x(1 - x)|x - t|, x ∈ [0, 1], t(t - 1) > 0, α > -1, β > -1, γ ∈ ℝ. As is shown in [D. Dai and L. Zhang, Painlevé VI and Henkel determinants for the generalized Jocobi weight, J. Phys. A: Math. Theor. 43 (2010), Article ID:055207, 14pp.], the corresponding Hankel determinant is the τ-function of a particular Painlevé VI. We present all the possible asymptotic expansions of the solution of the Painlevé VI equation near 0,∞ and 1 for generic (α,β,γ). For four special cases of (α,β,γ) which are related to the dimension of the Hankel determinant, we can find the exceptional solutions of the Painlevé VI equation according to the results of [A. Eremenko, A. Gabrielov and A. Hinkkanen, Exceptional solutions to the Painlevé VI equation, preprint (2016), arXiv:1602.04694], and thus give another characterization of the Hankel determinant.