Theta constant ($\Theta [\alpha ,\beta ](\tau )=\theta [\alpha ,\beta ](0,\tau )$)

A theta function with characteristics evaluated at spatial argument zero. Its dependence on $\tau \in \mathbb{H}$ remains holomorphic. Among the four half-integer characteristic choices, the odd one vanishes identically, whereas the three even constants are nonzero on the half-plane. The zeros of theta functions with characteristics locate their spatial zero away from the origin. Ratios of equal powers of even constants yield the modular lambda function and other modular-invariant functions.