TITLE: Do the complex zeros of the sum/difference of these series all reside on the line $\Re(s)=\frac12$?
QUESTION [9 upvotes]: The following series seems convergent for all $s\in \mathbb{C}$:
$$\displaystyle f(s):=\sum_{n=1}^\infty \frac{(-1)^n}{(n+s)^{n+s}}$$
The function itself does not appear to have any real or complex zeros, however numerical evidence suggests that all zeros of:
$$f(s) \pm f(1-s)$$
reside on the line $\Re(s)=\frac12$. Their density apparently increases in a quite regular manner. 

1) Could this be proven? When using the finite series $\sum_{n=1}^N$, the claim seems to hold for each $N$.
2) Could there exist a functional equation between $f(s)$ and $f(1-s)$? The only relation I found so far is that $f(-1)=f(1)$.
EDIT: A partial result on the second question. Based on Joro's findings below and realising that for integers and half-integers an infinite number of terms could be cancelled out when adding or substracting $f(s)$ and $f(1-s)$, the following reflective formulae hold:
For all integers $s \in \mathbb{Z}$:
$$f(s)+f(1-s) +\sum_{n=0}^{2(s-1)} \frac{(-1)^{n}}{(s-n)^{s-n}} =0$$ 
For all half-integers $s+\dfrac{1}{2}$ with $s \in \mathbb{Z}$:
$$f(s)-f(1-s) +\sum_{n=0}^{2(s-1)} \frac{(-1)^{n}}{(s-n)^{s-n}} =0$$
The non-alternating series $\displaystyle g(s):=\sum_{n=1}^\infty \frac{1}{(n+s)^{n+s}}$ has one formula for integers and half-integers:
$$g(s)-g(1-s) +\sum_{n=0}^{2(s-1)} \frac{1}{(s-n)^{s-n}} =0$$

REPLY [4 votes]: EDIT Due to confusing XRay and programming mistake, I erroneously claimed real zeros. Root finding didn't found any zeros off the line.

Very partial answer for functional equation.
For integer $s$, $\{1,f(s),f(1-s)\}$ appear linearly dependent
over the rationals with high precision.
Let $K=f(s),K_2=f(1-s),K_3=1/s^s,K_4=(K-K_2+K_3)^2$.
$$s=2, 1+4K+4K_2=0$$
$$s=3, 131-108K-108K_2=0$$
$$s=4, 36059+ 6912K+ 6912K_2=0$$
$$s=5, -695877463+ 21600000K+ 21600000K_2=0$$
$$s=6, 168087904001+ 583200000K+ 583200000K_2=0$$
$$s=7, -1639334733641495543+ 480290277600000K+ 480290277600000K_2=0$$
And in addition for $s=3/2, K^{2} - 2 \, K K_{2} + K_{2}^{2} + 2 \, K K_{3} - 2 \, K_{2} K_{3} + K_{3}^{2} - 2=0$.
...And with the help of Agno's observation, $s=5/2,8503056K_4^4 - 61096032K_4^3 + 202411224K_4^2 - 171663192K_4 + 252460321$
I suspect for $s$ half integer there is non-trivial algebraic dependency.