  Let $t$ be a positive integer and $\mathcal{A}$ a hereditary abelian category
satisfying some finiteness conditions. We define the semi-derived Ringel-Hall
algebra of $\mathcal{A}$ from the category
$\mathcal{C}_{\mathbb{Z}/t}(\mathcal{A})$ of $\mathbb{Z}/t$-graded complexes
and obtain a natural basis of the semi-derived Ringel-Hall algebra. Moreover,
we describe the semi-derived Ringel-Hall algebra by the generators and defining
relations. In particular, if $t$ is an odd integer, we show that there is an
embedding of derived Hall algebra of the odd-periodic relative derived category
in the extended semi-derived Ringel-Hall algebra.
