I'm trying to find a hypergeometric series for $ mathrm {Cl} 2 ( pi / 3) $, where

$$ mathrm {Cl} _2 (x) = – int_0 ^ x log left | 2 without frac {t} 2 right | dt = sum_ {k geq1} frac { without kx} {k ^ 2} $$

It is Clausen's function of order $ 2 $.

Context: Lately I have been very interested in the hypergeometric series, and I took them as a task to find a hypergeometric representation for the constant mentioned above.

What have I done.

We define $$ d_3 (n) = (- 1) ^ { left lfloor frac {n} 3 right rfloor} $$

as much as $$ chi_3 (n) = text {sgn} , text {mod} (n, 3) $$

With $$ text {sgn} , n = frac {n} {| n |} $$

Y $ text {sgn} , 0: = 0 $. So we have that

$$ mathrm {Cl} _2 left ( frac pi3 right) = frac { sqrt3} 2 sum_ {n geq1} d_3 (n) frac { chi_3 (n)} {n ^ 2 } $$

Why

$$ without frac {n pi} 3 = d_3 (n) chi_3 (n) frac { sqrt3} 2 $$

But I have no idea how to convert this into a hypergeometric series. Could I have some help? Thank you.