In this paper, we establish lower bounds for extreme values of derivatives of Dirichlet L-functions in the range $1/2<\sigma <1$. Compared with the work of Aistleitner et al. [‘On large values of $L(\sigma , \chi )$’, Q. J. Math. 70 (2019), 831–848], our result shows that derivatives of Dirichlet L-functions can attain extreme values of the same order of magnitude as the original L-functions in the interval ${1/2<\sigma <1}$.

In this paper, we establish a lower bound for the maxima of derivatives of the Dedekind zeta function of a cyclotomic field on the critical line. Employing a double-version convolution formula and combined with special GCD sums, our result generalizes the work of Bondarenko et al. [‘A dichotomy for extreme values of zeta and Dirichlet L-functions’, Bull. Lond. Math. Soc. 55 (2023), 2963–2975]. We also establish a lower bound via the resonance method when the real part is near the critical line; both of the above results refine parts of Yang’s [‘Extreme values of derivatives of the Riemann zeta function’, Mathematika 68 (2022), 486–510] work.

In this paper, we study the extreme values of the Rankin–Selberg L-functions associated with holomorphic cusp forms in the vertical direction. Assuming the generalised Riemann hypothesis (GRH), we prove that

$$ \begin{align*} \underset{T^{\delta}\leq t\leq T}{\max}\bigg\lvert L\bigg(\frac12+it,f\times f\bigg)\bigg\rvert \geq\exp\bigg(C\sqrt{\frac{\log T\log\log\log T}{\log\log T}}\bigg) \end{align*} $$

with $C\leq \mathscr {X}\sqrt {1-\delta }$, where $\mathscr {X}:=({2}/{\pi })\int _{0}^{\pi /3}\sin ^2\xi \,d\xi $ and $0\leq \delta <1$.

In this note, by introducing a new variant of the resonator function, we give an explicit version of the lower bound for $\log |L(\sigma ,\chi )|$ in the strip $1/2<\sigma <1$, which improves the result of Aistleitner et al. [‘On large values of $L(\sigma ,\chi )$’, Q. J. Math. 70 (2019), 831–848].