The first moment of primitive quartic L-functions over F_q(T) with fixed genus g equals q^{2g/3}(1-q^2)P(q^{-2})Z(q^{-2},q^{-1/2}) plus an error of size q^{(3/5+ε)g}.
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The first moment of central value of primitive quartic $L$-functions with fixed genus
The first moment of primitive quartic L-functions over F_q(T) with fixed genus g equals q^{2g/3}(1-q^2)P(q^{-2})Z(q^{-2},q^{-1/2}) plus an error of size q^{(3/5+ε)g}.