Regularized sums of half-integer powers of a Sturm–Liouville operator

We investigate the question of the regularized sums of part of the eigenvalues $z_n$ (lying along a direction) of a Sturm–Liouville operator. The first regularized sum is
$$\sum_{n=1}^\infty\left(z_n-n-\frac{c_1}n+\frac2\pi z_n\arctan\frac1{z_n}-\frac2\pi\right)=\frac{B_2}2-c_1\gamma+\int_1^\infty\left[R(\xi)-\frac{l_0}{\sqrt\xi}-\frac{l_1}\xi-\frac{l_2}{\xi\sqrt\xi}\right]\sqrt\xi\,d\xi,$$
where the $z_n$ are eigenvalues lying along the positive semi-axis, $z_n^2=\lambda_n$,
$$l_0=\frac\pi2,\quad l_1=-\frac12,\quad l_2=-\frac14\int_0^\pi q(x)\,dx,\quad c_1=-\frac2\pi l_2,$$
$B_2$ is a Bernoulli number, $\gamma$ is Euler's constant, and $R(\xi)$ is the trace of the resolvent of a Sturm–Liouville operator.