# Explicit formulae for L-function

1. Jun 1, 2009

### zetafunction

in a seminar at my univesity we had a brief introduction to prime number theorem , in the blackboard the professor wrote

$$\psi(x) =x-\sum_\rho\frac{x^\rho}{\rho} - \log(2\pi) -\log(1-x^{-2})/2$$

and he also gave an introduction (gave no proofs since we were in a seminar an were in a hurry) to the thing he called L-function

$$L(\chi,s)}=\sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}$$

and he said that 'Riemann zeta can be obtained from this ,

My question is , can we have an explicit formula in the form

$$? =x-\sum_\rho\frac{x^\rho}{\rho} - G(x)$$

but with the 'zeros' of the L-fucntions ? , i mean we should evaluate the contour integral

$$2i \pi f(x)= \int_{c-i\infty}^{c+i\infty}ds \frac{ L'(\chi,s)}{ L(\chi,s)}\frac{x^{s}}{s}$$