  We investigate the joint distribution of $L$-functions on the line $ \sigma=
\frac12 + \frac1{G(T)}$ and $ t \in [ T, 2T]$, where $ \log \log T \leq G(T)
\leq \frac{ \log T}{ ( \log \log T)^2 } $. We obtain an upper bound on the
discrepancy between the joint distribution of $L$-functions and that of their
random models. As an application we prove an asymptotic expansion of a
multi-dimensional version of Selberg's central limit theorem for $L$-functions
on $ \sigma= \frac12 + \frac1{G(T)}$ and $ t \in [ T, 2T]$, where $ ( \log
T)^\epsilon \leq G(T) \leq \frac{ \log T}{ ( \log \log T)^{2+\epsilon } } $ for
$ \epsilon > 0$.
