Geodesic
Asymptotical formula for fractional moments of some Dirichlet series
Čebyševskij sbornik, Tome 14 (2013) no. 1, pp. 18-33.

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Let $v \in \mathbf{N}$. Let the function $\Phi(T)$ arbitrarily slow tend to $+\infty$ with $T \rightarrow +\infty $. The asymptotical formulas for fractional moments of the Riemann zeta-function $\int\limits_T^{2T}|\zeta(\sigma+it)|^{2/v}dt$ for ${1}/{2}+{\Phi(T)}/{\ln T}\le \sigma1$ and for fractional moments of the arithmetical Dirichlet series of second degree from Selberg's class $\int\limits_T^{2T}|L(\sigma+it)|^{2/v}dt$ for ${1}/{2}+{\Phi(T)}/{\sqrt{\ln T}}\le \sigma1$, are obtained. 
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S. A. Gritsenko; L. N. Kurtova. Asymptotical formula for fractional moments of some Dirichlet series. Čebyševskij sbornik, Tome 14 (2013) no. 1, pp. 18-33. http://geodesic.mathdoc.fr/item/CHEB_2013_14_1_a2/

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