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We study the distribution of large (and small) values of several families of$L$-functions on a line $\text{Re(s)}=\sigma$ where $1/2<\sigma<1$. We considerthe Riemann zeta function $\zeta(s)$ in the $t$-aspect, Dirichlet $L$-functionsin the $q$-aspect, and $L$-functions attached to primitive holomorphic cuspforms of weight $2$ in the level aspect. For each family we show that the$L$-values can be very well modeled by an adequate random Euler product,uniformly in a wide range. We also prove new $\Omega$-results for quadraticDirichlet $L$-functions (predicted to be best possible by the probabilisticmodel) conditionally on GRH, and other results related to large moments of$\zeta(\sigma+it)$.

On the distribution of extreme values of zeta and $L$-functions in the strip $1/2<\sigma<1$