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When the matrix size $n\\to\\infty$, we obtain an asymptotic formula for the Hankel determinants, valid uniformly for $t\\in (0, d]$, $d>0$ fixed. A particular Painlev\\'{e} III transcendent is involved in the approximation, as well as in the large-$n$ asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. 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