I've been looking at the zeros of the incomplete zeta function
$\zeta_{lower}(s, z)$ recently.
$$
\zeta_{\mathrm{lower}}(s,z)=-\frac{{\Gamma(1-s)}}{2\pi i}\int_{z}^{\infty}\frac{{(-t)^{s-1}}}{e^{t}-1}dt
$$
The subscript "lower" indicates that the contour of integration passes below
the branch cut on the positive real axis.
By zeros, I mean I mean the zeros in $z$, for fixed $s$.
I find that this function can have
an infinite number of complex zeros $z_k$, each of which is a function of the parameter $s$. The zeros (for fixed $s$) live on a smooth curve and have an accumulation point at the origin. 

Let $s_0$ be any zero of the (original, complete) Riemann zeta function.  A remarkable
thing happens when $s$ is near $s_0$.  The kth zero, $z_k(s)$ depends
logarithmically on $(s_0-s)$.
$$
\log(z_k) \approx \frac{1}{s-1} \log(s_0-s) + C
$$
Assuming $|\operatorname{Im}(s)| >> |\operatorname{Re}(s)|$, the entire set $z_k$ rotate about the origin as
$s \rightarrow s_0$.  Imagine a (slightly bent) barn door rotating about its hinge. 

For added fun, if you move $s$ around $s_0$ and return to the original point, $z_k$ will become $z_{k+1}$ or $z_{k-1}$ depending on the sense of the rotation in $s$.

I've verified these prediction numerically.

So my question is whether anyone has seen this before, for the zeta
function or for other incomplete functions.