Integrals of L-series for automorphic forms on hyperbolic three-space are studied. Let F be an automorphic form for the group \(SL_ 2({\mathfrak O})\) with \({\mathfrak O}\) the ring of integers of \({\mathbb{Q}}(\sqrt{-1})\), let \(L_ f(s,m)\) be its L-series, twisted by the character \(\alpha \mapsto (\alpha /| \alpha |)^{4m}\). It is shown that \[ \sum_{| m| \leq T}\int^{T}_{-T}| L_ f(+it,m)|^ 2 dt\ll T^ 2(\log T)^ b \] with \(b=1\) if f is a cusp form and \(b=4\) otherwise. By taking for f a suitable derivative of the Eisenstein series the author obtains the corresponding result for the fourth power of \[ L(s,\lambda^ m)=\sum_{\alpha \neq 0,\alpha \in {\mathfrak O}}(\alpha /| \alpha |)^{4m} N(\alpha)^{-s}. \]