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 $\sigma= \frac12 + ( \log T)^{-\theta}$ and $t \in [ T, 2T]$, where $0 < \theta < \frac12$ is a constant.