explicit formulae for L-function

**zetafunction** · Jun1-09, 05:53 AM

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

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

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

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

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

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

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

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

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