Conditional estimates for $L$-functions in the Selberg class

Assuming the Generalized Riemann Hypothesis, we provide uniform upper bounds with explicit main terms for moduli of $\left({\text{cL}}^{\mathrm{\prime }}\mathrm{/}\text{cL}\right)(s)$ and $\log \text{cL}(s)$ for $1\mathrm{/}2+\delta \le \sigma <1$, fixed $\delta \in (0,1\mathrm{/}2)$ and for functions in the Selberg class except for the identity function. We also provide estimates under additional assumptions on the distribution of Dirichlet coefficients of $\text{cL}(s)$ on prime numbers. Moreover, by assuming a polynomial Euler product representation for $\text{cL}(s)$, we establish uniform bounds for $\mathrm{\mid }3\mathrm{/}4-\sigma \mathrm{\mid }\le 1\mathrm{/}4-1\mathrm{/}\log \log \left(\text{sq}\mathrm{\mid }t{\mathrm{\mid }}^{\text{sdeg}}\right)$, $\mathrm{\mid }1-\sigma \mathrm{\mid }\le 1\mathrm{/}\log \log \left(\text{sq}\mathrm{\mid }t{\mathrm{\mid }}^{\text{sdeg}}\right)$ and $\sigma =1$, and completely explicit estimates by assuming also the strong $\lambda$-conjecture.