Clearly, we have that $f(z) = z + \sum_{k=1}^{\infty} a_k z^{3k+1}$. Incidentally, it turns out that the function $g(z) = \sin(\pi z)e^{\pi z/\sqrt{3}}$ has exactly the same zeros as $f(z)$ [note that $g(z)$ has "periodic" derivatives $g'''(z) = -\alpha^3g(z)$ with $\alpha = 2\pi/\sqrt{3}$].
For this reason, I speculate that:
$$f(z) = \sin(\pi z)\exp\left(\cfrac{\pi z}{\sqrt{3}}\right) \exp(\phi(z))$$
for some unknow function $\phi(z)$. Would it be possible to determine easily a closed form for $\phi(z)$?
*Note that if $\phi(z)$ were known, the irrational value of $\zeta(3)$ could be determined in a closed form (which seems complicated in the light of Roger Apéry's work):
$$\zeta(3) = -a_1 = \lim_{z\to 0} -\frac{1}{3!}\frac{d(f(z)/z)}{dz} = -3\pi\left[\phi''(0) + \frac{2\pi}{\sqrt{3}}\phi'(0)\right]$$