Estimates of the form
$L^{(j)}(s,\mathscr{A})\ll_{\epsilon,j,\mathscr {D_A}}\mathscr
{R}^\epsilon_{\mathscr {A}}$ in the range $|s-1|\ll 1/\log \mathscr
{R_A}$ for general $L$-functions, where $\mathscr {R_A}$ is a
parameter related to the functional equation of $L(s,\mathscr {A})$,
can be quite easily obtained if the Ramanujan hypothesis is
assumed. We prove the same estimates when the $L$-functions have Euler
product of polynomial type and the Ramanujan hypothesis is replaced by
a much weaker assumption about the growth of certain elementary
symmetrical functions. As a consequence, we obtain an upper bound of
this type for every $L(s, \pi)$, where $\pi$ is an automorphic cusp
form on ${\rm GL}(\mathbf {d},\mathbb {A}_K)$. We employ these results
to obtain Siegel-type lower bounds for twists by Dirichlet characters
of the third symmetric power of a Maass form.