My question refers to article 301 of section 5 of Gauss's D. A. - there Gauss gives an asymptotic formula for the mean number of classes of forms with positive discriminant ($D>0$):

$$h(D) = \frac{4}{\pi^2}\log (D) + \delta$$

where $\delta$ is the following constant:

$$\delta = \frac{8}{\pi^2}C+\frac{48}{\pi^4}\sum_{n=2}^{\infty}\frac{\log (n)}{n^2} - \frac{2 \log 2}{3\pi^2}$$

and $C$ is the Euler-Mascheroni constant. This formula is noteworthy because Gauss also evaluates the error term $\delta$ in this class formula. I tried to search on the web for references to this analytic result of Gauss but all I found is articles about his asymptotic class number formula for forms with negative discriminant ($D<0$), which Gauss gives in the next article of D. A. (article 302).

My question is mainly about getting a reference for this result of Gauss. If such a reference doesn't exist, I'll be glad to hear an expert opinion on the historic significance of this formula (Gauss did publish this formula; thereby it's plausible that it did have influence).