The Hurwitz zeta function:

$$\zeta_{H}(s,a)$$

reduces to $\zeta(s)$ when $a=1$ and to $(2^s-1)\zeta(s)$ when $a=\frac12$.

However, I stumbled upon a peculiar third connection:

$$\zeta_{H}(s,a) + \zeta_{H}(s,1-a)$$

that seems to exactly produce the non-trivial zeros of $\zeta(s)$,

when $a=\frac12$ (obviously), but also (and apparently only) when $a=\frac13, \frac14$ or $\frac16.$

Why does it only work for these values? Is there any reference to this in the literature?

Thanks.