Too long for a comment.
After @Martin Hansen's answer, I had a look to he zero of equation
$$f_n(x)=\sum_{k=1}^{n} k^x$$
(give a lot of attention to @reuns's comment).
I used Newton method. For the base case $(n=3)$ starting with $x_0=-1+\pi\,i$ the iterates are
$$\left(
\begin{array}{cc}
1 & -0.501820 \,+ \,3.86512 \,i \\
2 & -0.487049 \,+ \,3.58176\,i \\
3 & -0.454116 \,+ \,3.59875 \,i \\
4 & -0.454397 \,+ \,3.59817\, i
\end{array}
\right)$$
Below are given some results which could be a good start (I hope) for a deeper exploration (as @Martin Hansen suggested).
$$\left(
\begin{array}{cc}
3 & -0.454397\, + \,3.59817 \, i \\
4 & -0.625971\, + \,3.12712 \, i \\
5 & -0.714285\, + \,2.83349 \, i \\
6 & -0.767633\, + \,2.62901 \, i \\
7 & -0.803209\, + \,2.47644 \, i \\
8 & -0.828584\, + \,2.35711 \, i \\
9 & -0.847585\, + \,2.26049 \, i \\
10 & -0.862348\, + \,2.18022 \, i \\
11 & -0.874153\, + \,2.11212 \, i \\
12 & -0.883812\, + \,2.05341 \, i \\
13 & -0.891868\, + \,2.00209 \, i \\
14 & -0.898692\, + \,1.95672 \, i \\
15 & -0.904551\, + \,1.91623 \, i \\
16 & -0.909640\, + \,1.87978 \, i \\
17 & -0.914103\, + \,1.84675 \, i \\
18 & -0.918051\, + \,1.81661 \, i \\
19 & -0.921571\, + \,1.78897 \, i \\
20 & -0.924730\, + \,1.76349 \, i
\end{array}
\right)$$