This is a follow up question to this one.

It seems that results on moments of L-functions, that is, estimates for integrals of the form $$\int^{T}_1|\zeta(\sigma+it)|^{2k}dt$$ are typically for the incomplete L-function itself. Are there results for moments of completed L-functions, say, the completed Riemann zeta function?

If not, is the reason simply because the completed expression is more complicated than the incomplete one?