Let s: [1,∞) → ℂ be a locally Lebesgue integrable function. We say that s is summable (L,1) if there exists some A ∈ ℂ such that $li{m}_{t\to \infty }\tau \left(t\right)=A$, where $\tau \left(t\right):=1/\left(logt\right){\int }_1^{t}s\left(u\right)/udu$. (*) It is clear that if the ordinary limit s(t) → A exists, then also τ(t) → A as t → ∞. We present sufficient conditions, which are also necessary, in order that the converse implication hold true. As corollaries, we obtain so-called Tauberian theorems which are analogous to those known in the case of summability (C,1). For example, if the function s is slowly...