Here is a very interesting continued fraction expansion for pi/2 :
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![{\displaystyle \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}})}}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/dbe75042d22ec00bb32b3e38a5d706ff7050b136)
It appears to converge extremely slowly.
![{\displaystyle \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}})}}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/dbe75042d22ec00bb32b3e38a5d706ff7050b136)
