  In this paper, we prove that $\zeta$ is not a solution of any non-trivial
algebraic differential equation whose coefficients are polynomials in $\Gamma,
\Gamma^{(n)}, \Gamma^{(l)}$ over the ring of polynomials in $\mathbb{C}$,
$l>n\geq 1$ are positive integers. We extended the result that $\zeta$ does not
satisfy any non-trivial algebraic differential equation whose coefficients are
polynomials in $\Gamma, \Gamma', \Gamma''$ over the field of complex numbers,
which is proved by Li and Ye[7].
