For monic square-free polynomials of even degree over F_q[x], the first moment of L(1/2, chi_D) has secondary main terms of sizes q^((2g+2)/3), q^(g/6+floor(g/2)) and q^(g/6+floor((g-1)/2)), with error O(q^(g(1+eps)/2)).
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The First Moment of $L(\frac{1}{2},\chi)$ for Real Quadratic Function Fields
For monic square-free polynomials of even degree over F_q[x], the first moment of L(1/2, chi_D) has secondary main terms of sizes q^((2g+2)/3), q^(g/6+floor(g/2)) and q^(g/6+floor((g-1)/2)), with error O(q^(g(1+eps)/2)).