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The first moment of central value of primitive quartic $L$-functions with fixed genus
We investigate the mean value of the first moment of primitive quartic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{χ primitive\ quartic\\ χ^2 primitive\\ genus(χ)=g}}L_q(\frac{1}{2}, χ), \end{equation*} where
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