With $s \in \mathbb{C}, a \in \mathbb{R}$, 

numerical evidence strongly suggests that the complex zeros in the critical strip of:

$$\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$$

all reside on the line $\Re(s)=\frac12$ for $a \lt 0$ or $a\ge 1$. There also exist a finite few complex zeros for each $a$, but these are outside the critical strip. 

When $a=1$ the formula reduces to [this question][1].

**Question:** Is there an explanation for this phenomenon (or a counter example)?


  [1]: https://mathoverflow.net/questions/89518/are-the-semi-trivial-zeros-of-zetas-pm-zeta1-s-all-on-the-critical-li