The formula some simple zeros of the Dirichlet eta function

Question

Let $s$ be a complex variable with $\Re(s)>0$. The Dirichlet eta function $\eta(s)$ is defined by $$\eta(s)=(1-2^{1-s})\zeta(s)$$ where $\zeta(s)$, of course, is the Riemann zeta function. We know that the factor $(1-2^{1-s})$ has infinitely many zeros on the critical line $\Re(s)=1$. Specifically, the zeros of this factor take the form $$s_n=1+n\frac{2\pi i}{\log2}$$ where $n\in\mathbb{Z}$. If we set $(1-2^{1-s})=0$, the only trivial solution is $s=1$. How does one obtain the general solution $s_n$ above for the other zeros to include the imaginary part?