Analytical continuation of $-\frac{t^w}{\ln(2)}\sum_{k=-\infty}^\infty \frac{t^{w k}}{e^{\pi i w(1+2k)}-1}$

Question

I'm interested in obtaining the analytical continuation of the following function. Let $w = \frac{2 \pi i}{\ln(2)}$, then let $$F(t) = -\frac{t^{w/2}}{\ln(2)}\sum_{k=-\infty}^\infty \frac{t^{w k}}{e^{\pi i w(1+2k)}-1}$$ I think it might be the case that this function has a natural boundary on the line $\mathfrak{Re}(t) = 0$, however, a formula for $F(t)$ that converges on both sides of this boundary would be especially helpful.