  We show the estimates \inf_T \int_T^{T+\delta} |\zeta(1+it)|^{-1} dt
=e^{-\gamma}/4 \delta^2+ O(\delta^4) and \inf_T \int_T^{T+\delta} |\zeta(1+it)|
dt =e^{-\gamma} \pi^2/24 \delta^2+ O(\delta^4) as well as corresponding results
for sup-norm, L^p-norm and other zeta-functions such as the Dirichlet
L-functions and certain Rankin-Selberg L-functions. This improves on previous
work of Balasubramanian and Ramachandra for small values of \delta and we
remark that it implies that the zeta-function is not universal on the line
Re(s)=1. We also use recent results of Holowinsky (for Maass wave forms) and
Taylor et al. (Sato-Tate for holomorphic cusp forms) to prove lower bounds for
the corresponding integral with the Riemann zeta-function replaced with Hecke
L-functions and with \delta^2 replaced by \delta^{11/12+\epsilon} and
\delta^{8/(3 \pi)+\epsilon} respectively.
