$$\frac{1}{2\pi i}\int_{I(\lambda,\infty)}^{} \left(\frac{1}{t}+\frac{1}{2}+\frac{1}{12}t\right)\frac{e^{-zt}\log t}{t^2}\,dt$$ where $I(\lambda, \infty)$ is the integral path consisting of $(\infty, \lambda)$ , counterclockwise circle of radius $\lambda$ around the origin and $(\lambda, +\infty)$
Please teach me how to calculate this contour integrate.