Legendre chi in terms of Lerch transcendent
From specialfunctionswiki
Theorem
The following formula holds: $$\chi_{\nu}(z)=2^{-\nu} z \Phi \left(z^2,\nu,\dfrac{1}{2} \right),$$ where $\chi$ denotes Legendre chi and $\Phi$ denotes the Lerch transcendent.