  We discuss the life span of the Cauchy problem for the one-dimensional
Schr\"{o}dinger equation with a single power nonlinearity $\lambda |u|^{p-1}u$
($\lambda\in\mathbb{C}$, $2\le p<3$) prescribed an initial data of the form
$\varepsilon\varphi$. Here, $\varepsilon$ stands for the size of the data. It
is not difficult to see that the life span $T(\varepsilon)$ is estimated by
$C_0 \varepsilon^{-2(p-1)/(3-p)}$ from below, provided $\varepsilon$ is
sufficiently small. In this paper, we consider a more precise estimate for
$T(\varepsilon)$ and we prove that $\liminf_{\varepsilon\to
0}\varepsilon^{2(p-1)/(3-p)}T(\varepsilon)$ is larger than some positive
constant expressed only by $p$, $\mathrm{Im}\lambda$ and $\varphi$.
