Zeros of derivatives of Dirichlet Eta function

Let $$\eta^{(d)}(z) = \sum_{n=1}^\infty \dfrac {(-1)^d(-1)^{n-1}\ln(n)^d} {n^z}$$ be the derivative of Dirichlet Eta function of order $$d$$.

Does it exist any known or not known zero of $$\eta^{(d)}(z)$$ such that $$d \geq 1$$ and $$\frac{1}{2}<\Re(z)<1$$

Thanks

Not an answer, but relevant: Bohr and Landau showed in [1] that if a Dirichlet series converges for $$\sigma>0$$, then $$N(\alpha,T)$$, the number of zeros for $$\sigma>\alpha$$ and $$0, is $$O(T)$$ for $$\alpha>1/2$$.

Update ($$d=1$$ case): The Mathematica command

FindRoot[2^(1 - s) Log[2] Zeta[s] + (1 - 2^(1 - s)) Zeta'[s], {s,1. + 95. I}]

returns {s -> 0.926336 + 95.3143 I}

[1] Ein Satz über Dirichletsche Reihen mit Anwendung auf die $$\zeta$$ Funktion und die $$L$$-Funktionen, Rend. Circ. Mat. Palermo 37 (1914), 269-272.

• Thanks @Stopple for your suggestion, so we don't know if $\eta^{(d)}(z)=0$ when $d \geq 1$ and $\frac{1}{2} < \Re(z) < 1$ ? – Matey Math Feb 10 at 23:58
• thanks @Stopple for your example – Matey Math Feb 13 at 13:41