  In this paper, we determine the order of magnitude of the $2q$-th
pseudomoment of powers of the Riemann zeta function $\zeta(s)^{\alpha}$ for
$0<q\le 1/2$ and $0< \alpha<1$, completing the results of Bondarenko, Heap and
Seip, and of Gerspach. Our results also apply to more general Euler products
satisfying certain conditions.
