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Let $l=-1/2+i\,t$; $t\in\mathbb{R}$. I'm wondering if there is a way to compute the following integral $$\int_{-1}^xP^m_l(\gamma)^2 \,d\,\gamma; -1<x<1.$$ In which $P^m_l$ is the Legendre's first kind function of degree $l=-1/2+it$ and of order $m\in\mathbb{N}$.

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