= Solution
For the intended complex unit circle, put $u=\alpha(1)$ and $v=\alpha(i)$, with $|u|=|v|=1$. The complete family of <semicharacters of a complex lattice> is
$$
\boxed{\alpha(m+ni)=(-1)^{mn}u^m v^n.}
$$
To obtain this, the identity at zero gives $\alpha(0)=1$; on each coordinate axis the alternating form vanishes, so $\alpha(m)=u^m$ and $\alpha(ni)=v^n$, also for negative integers. Combining the axes gives the factor $(-1)^{mn}$. Conversely, for $\lambda=m+ni$, $\mu=r+si$, the quotient of the proposed values is $(-1)^{ms+nr}$, which equals $e^{i\pi(nr-ms)}$. Hence every displayed formula is a <semicharacter of a complex lattice> and none are missing.
The original PDF actually puts $z\in\mathbb Z$ in its definition of $C_1$. Under that literal reading $C_1=\{1,-1\}$, so \b[there are exactly four solutions], given by the same formula with $u,v\in\{1,-1\}$. Under either reading the specified choice is $\alpha_1(m+ni)=(-1)^{mn}$.
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