Power moments of automorphic L-functions related to Maass forms for SL 3(ℤ)

Let f f be a self-dual Hecke-Maass eigenform for the group S L 3 ( Z ) S{L}_{3}\left({\mathbb{Z}}) . For 1 2 < σ < 1 \frac{1}{2}\lt \sigma \lt 1 fixed we define m ( σ ) m\left(\sigma ) ( ≥ 2 \ge 2 ) as the supremum of all numbers m m such that ∫ 1 T ∣ L ( s , f ) ∣ m d t ≪ f , ε T 1 + ε , \underset{1}{\overset{T}{\int }}| L\left(s,f){| }^{m}{\rm{d}}t{\ll }_{f,\varepsilon }{T}^{1+\varepsilon }, where L ( s , f ) L\left(s,f) is the Godement-Jacquet L-function related to f f . In this paper, we first show the lower bound of m ( σ ) m\left(\sigma ) for 2 3 < σ < 1 \frac{2}{3}\lt \sigma \lt 1 . Then we establish asymptotic formulas for the second, fourth and sixth powers of L ( s , f ) L\left(s,f) as applications.