  The zero divisor of the theta function of a compact Riemann surface $X$ of
genus $g$ is the canonical theta divisor of Pic${}^{(g-1)}$ up to translation
by the Riemann constant $\Delta$ for a base point $P$ of $X$. The complement of
the Weierstrass gaps at the base point $P$ given as a numerical semigroup plays
an important role, which is called the Weierstrass semigroup. It is classically
known that the Riemann constant $\Delta$ is a half period
$\frac{1}{2}\Gamma_\tau$ for the Jacobi variety
$\mathcal{J}(X)=\mathbb{C}^g/\Gamma_\tau$ of $X$ if and only if the Weierstrass
semigroup at $P$ is symmetric. In this article, we analyze the non-symmetric
case. Using a semi-canonical divisor $D_0$, we show a relation between the
Riemann constant $\Delta$ and a half period $\frac{1}{2}\Gamma_\tau$ of the
non-symmetric case. We also identify the semi-canonical divisor $D_0$ for
trigonal curves, and remark on an algebraic expression for the Jacobi inversion
problem using the relation
