Also, to give an idea of what I am going to do:
\[
\frac{d^{s}}{dw^s}\Big{|}_{w=0} \frac{e^{iw} + e^{-iw}}{2} -1 = \frac{i^s + (-i)^s}{2}= \cos(\frac{\pi s}{2})\,\,\text{when}\,\,0 < \Re(s) < 1
\]
Who's to say the same thing doesn't happen with \(\vartheta_\pi\)
\[
\frac{d^{s}}{dw^s}\Big{|}_{w=0} \frac{e^{iw} + e^{-iw}}{2} -1 = \frac{i^s + (-i)^s}{2}= \cos(\frac{\pi s}{2})\,\,\text{when}\,\,0 < \Re(s) < 1
\]
Who's to say the same thing doesn't happen with \(\vartheta_\pi\)