TITLE: Why is it that this gives a good approximation of $\pi$?
QUESTION [30 upvotes]: At the end of a Physics examination, I decided to play around with my calculator, as I always do when I have time left over, and found that:
$$\frac{1}{100} \cdot 11^{\ln(11)} \approx 3.14159789211,$$ where $\ln(x)$ is the natural logarithm of $x$, gives $\pi$ correct to five decimal places, where $\pi \approx 3.14159265359$.
Does anybody know of any reason why this may be, or if this is simply a coincidence?
Edit
I must thank @Shailesh for providing me with the link to the following Reddit page, for it also begs the same question as to whether or not there is a relationship between $11, \ln(11),$ and $\pi$.

REPLY [2 votes]: We can write the approximation as the following equivalent almost-integer.
$$\frac{11^{\log(11)}}{\pi}\approx 100.0001667\approx 10^2+\frac{1}{6000}$$
Two similar ones with three zeros after the decimal point also involve multiples of Heegner numbers.
$$\frac{(4\times43)^{\log(4\times43)}}{\pi}\approx 102381257746.0007223$$
$$\frac{(20\times163)^{\log(20\times163)}}{\pi} \approx 8374566425664827543971729687.0002623$$
Yet another Heegner number, $19$, appears in the almost-integer
$$\frac{2(2\times 19)^{\log(2\times 19)}}{\pi} \approx  355204.9991 \approx5\left(41+71\times10^3\right)=4^2(12^2+5)^2-11$$
which gives an approximation to $\pi$ with seven correct decimal digits.
$$\pi \approx \frac{2\times38^{\log(38)}}{355205} \approx 3.1415926$$