A limit involving the Thue–Morse sequence

(it is a follow-up on my previous question)

Let's define $t_n$ by the recurrence $$t_0 = 1, \quad t_n = (-1)^n \, t_{\lfloor n/2\rfloor}.\tag1$$ It is easy to see that $|t_n|=1$, and the signs follow the same pattern as the Thue–Morse sequence: $$1,\,-1,\,-1,\,1,\,-1,\,1,\,1,\,-1,\,-1,\,1,\,1,\,-1,\,1,\,-1,\,-1,\,1,\,...\tag2$$ (see this question for an example of a non-recursive formula for $t_n$).

Now, let: $$\mathcal{L}(z)=\lim_{n\to\infty}\,\sum_{k=1}^{2^n-1}t_k\,k^z.\tag3$$ I conjecture that the following propositions hold:

$\color{gray}{\text{(a)}}$ The limit $\mathcal{L}(z)$ exists for all $z\in\mathbb C$.

$\color{gray}{\text{(b)}}$ $\mathcal{L}(z)$ is an entire function of $z$.

$\color{gray}{\text{(c)}}$ $\mathcal{L}(z)=0$ if and only if $z\in\mathbb Z^+$.

Can we prove these conjectures? Can we find a different representation of $\mathcal{L}(z)$? Does this function have any interesting properties?

Update: The "if" part of the conjecture $\color{gray}{\text{(c)}}$ is certainly true. I am now pretty sure the "only if" part fails at some points on the imaginary axis (e.g. near $z\approx i\,4.53236...$; is it exactly $i\,\pi/\ln2$?). I am still wondering if there are any zeros except positive integers and off the imaginary axis.