I wonder if there is a closed formula for the following generalized hypergeometric function at $z=1$:

$${}_4F_3\left(a,a,a,a;a+1,a+1,2a;1\right)$$

Specifically, I would like to have a formula in terms of gamma functions and its derivatives, analogous to the identity

$${}_3F_2\left(a,a,a;a+1,2a;1\right) = \frac{\Gamma(2a+1)}{4\Gamma(a)^2}\left[\psi'\left(\frac{a}{2}\right)-\psi'\left(\frac{a+1}{2}\right)\right]\,,$$

where $\psi(z)=\Gamma'(z)/\Gamma(z)$.