Alanonala

$\varphi \left(e^{-6\pi }\right)={\frac {{\sqrt[{3}]{3{\sqrt {2}}+3{\sqrt[{4}]{3}}+2{\sqrt {3}}-{\sqrt[{4}]{27}}+{\sqrt[{4}]{1728}}-4}}\cdot {\sqrt[{8}]{243{\pi }^{2}}}}{6{\sqrt[{6}]{1+{\sqrt {6}}-{\sqrt {2}}-{\sqrt {3}}}}{\Gamma ({\frac {3}{4}})}}}$

$\sum _{n=1}^{\infty }{\frac {2}{n(5n-3)}}={\frac {1}{15}}{\pi }{\sqrt {25-10{\sqrt {5}}}}+{\frac {2}{3}}\ln(5)+{\frac {{1}+{\sqrt {5}}}{3}}\ln \left({\frac {1}{2}}{\sqrt {10-2{\sqrt {5}}}}\right)+{\frac {{1}-{\sqrt {5}}}{3}}\ln \left({\frac {1}{2}}{\sqrt {10+2{\sqrt {5}}}}\right)$

$\sum _{n=1}^{\infty }{\frac {2}{n(5n-3)}}=$

Here is a very interesting continued fraction expansion for pi/2 :

$\pi /2=1+{\cfrac {1}{1+{\cfrac {1}{1/2+{\cfrac {1}{1/3+{\cfrac {1}{1/4+{\cfrac {1}{1/5+{\cfrac {1}{1/6+{\cfrac {1}{1/7+{\cfrac {1}{1/8+{\cfrac {1}{1/9+{\cfrac {1}{1/10+{\cfrac {1}{1/11+{\cfrac {1}{1/12+{\cfrac {1}{1/13+{\cfrac {1}{1/14+{\cfrac {1}{1/15+\ddots }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

It appears to converge extremely slowly.