As it says in the title - I'm looking for results on \[ \int _1^T|\zeta ^{(n)}(\sigma +it)|^{2k}dt\] for general $n\in \mathbb N$. Are there any? I only need upper bounds and logarithm factors aren't important.
I'm aware that (at least the lower) moments are worked out using the approximate functional equation and consequently the proofs would go through for derivatives as well, the sums involved only changing by logarithm factors.