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A recent generalization of the "Kleinian sigma function" involves the choiceof a point $P$ of a Riemann surface $X$, namely a "pointed curve" $(X, P)$.This paper concludes our explicit calculation of the sigma function for curvescyclic trigonal at $P$. We exhibit the Riemann constant for a Weierstrasssemigroup at $P$ with minimal set of generators $\{3, 2r+s,2s+r\}$, $r<s$,equivalently, non-symmetric, we construct a basis of $H^1(X, \mathbb{C})$ and afundamental 2-differential on $X\times X$, we give the order of vanishing forsigma on Wirtinger strata of the Jacobian of $X$, and a solution to the Jacobiinversion problem.

The sigma function for trigonal cyclic curves