Mixed integral moments of the Hecke $L$-functions and Riemann zeta function

In this paper, let $f$ be a Hecke cusp form for $SL(2,\mathbb{Z})$. We establish an asymptotic formula for the mixed moment of ${\zeta }^2(s)$ and $L(s,f)$ on the critical line, valid for both holomorphic and Maass forms.