Are the complex zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$ all on the critical line for $a \lt 0, a \ge 1$?

Question

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.

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