Asymptotical formula for fractional moments of some Dirichlet series

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 ${1}/{2}+{\Phi(T)}/{\ln T}\le \sigma<1$ 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 ${1}/{2}+{\Phi(T)}/{\sqrt{\ln T}}\le \sigma<1$, are obtained.