The representation zeta function of the first principal congruence subgroup of \operatorname{SL}_2$\mathbb{Z}_2$

Let $\operatorname{SL}_2^1(\mathbb{Z}_2)$ denote the first principal congruence subgroup of $\operatorname{SL}_2(\mathbb{Z}_2)$, and let $\zeta_{\operatorname{SL}_2^1(\mathbb{Z}_2)}(s)$ be its representation zeta function.

The representation zeta function conjecture. The zeta function is

$$\zeta_{\operatorname{SL}_2^1(\mathbb{Z}_2)}(s) = \frac{2^{5}(2^2 - 2^{-s})}{1 - 2^{1-s}}.$$

This formula is suggested by computer-aided calculations and extends the explicit formula obtained in the surrounding computation to the exceptional case $p=2$. The source provides no proof or resolution of this specific assertion.