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



Zeros of derivatives of Dirichlet Eta function
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 $0<t<T$, 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.


