Numerical evaluation of this series

zetafunction

let be the series [tex] \sum_{n=0}^{\infty}\Lambda (n) cos(ulogn) [/tex] her /\(n) is the Mangoldt function.

i know this series is divergent for any value of parameter 'u' , my question is if there is any numerical method to obtain a finite value from this series by truncating upto some finite N

the zeta-regularized value of this series is the Real part of [tex] 1/2 \frac{d}{du}log \zeta (1/2+iu) [/tex] but i would like to know if there is a method to evaluate it numerically.