Riemann's functional equation may be written: $$ \frac{\zeta(s)}{\zeta(1-s)} = 2^s \pi^{s-1} \sin(\frac{\pi s}2) \Gamma(1-s) \tag{1} $$ and so by symmetry: $$ \frac{\zeta(1-s)}{\zeta(s)} = 2^{1-s} \pi^{-s} \cos(\frac{\pi s}2) \Gamma(s) \tag{2} $$ multiplying the two versions gives Euler's reflection formula.
now define a function $$ \Psi(s) = (2\pi)^{-s} \Gamma(s) \zeta^2(s) $$ so that the result of dividing $(1)$ by $(2)$ is expressed as: $$ \frac{\Psi(s)}{\Psi(1-s)} = \tan \frac{\pi s}2 $$
question is there any intuitively appealing "explanation" for this periodicity?