Selberg's central limit theorem of $L$-functions near the critical line

We find an asymptotic expansion of a multi-dimensional version of Selberg’s central limit theorem for $L$-functions on $\sigma =\frac{1}{2}+(\log T{)}^{-\theta }$ and $t\in [T,2T]$, where $0<\theta <\frac{1}{2}$ is a constant.