 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 isTo 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 there are exactly four solutions, given by the same formula with $u,v\in \{1,-1\}$. Under either reading the specified choice is ${\alpha }_1(m+ni)=(-1{)}^{mn}$.