I am working on an enumerative problem related to knot theory, and I have found the following generating function

$$F(z)=\prod_{n\geq 1} \frac{1}{(1-z^{2n+1})^2}.$$ I am interested on getting asymptotic estimates for this GF. I am not an expert on partition function theory, but it seems that trying to apply Meinardus Theorem on this context should be a good idea (in this case, the sequence of coefficients to be studied is $a_n=2$ for $n=2k+1$ and $\geq 3,$ and $a_n=0$ in the rest of the cases.

Once trying to apply Meinardus Theorem (as it is stated in the reference book of George Andrews: The theory of partitions) I have seriuos troubles with condition III, namely that

$$f(t,y)=Re(g(e^{-t-2\pi i y}))-g(e^{-t})\leq -C t^{-\varepsilon}$$

for $g(z)=\sum_{n\geq 1} a_n z^n$, $|y|\leq 1/2$, $t$ small enough and convenient choices of positive $C$ and $\varepsilon$. Essentially, the problem I am finding is that when taking $y=1/2$ and $t$ small (or in a neighbourhood), the value I get for $f(t,y)$ is $-2t+o(t)$, which cannot fit with the condition $\leq -C t^{-\varepsilon}$.

So: which are the alternatives that one can use to get the assymptotics in such a situation (without having to make the computation of the contour integral computation?)

I was thinking also to try to get results exploiting the asymptotics of the partition function, but I do not see a direct way.