The classical explicit formula for the Riemann Zeta function states that
$$
\psi(x)=x-\sum\_{\rho} \frac{x^{\rho}}{\rho}+O(1), 
$$
where $\psi(x)=\sum\_{n \leq x} \Lambda(n)$ and the sum is over all non-trivial zeroes of $\zeta(s)$.


Let $L/K$ be a finite Galois extension of number fields and let $V$ be an irreducible complex representation of $\textrm{Gal}(L/K)$. Attached to this piece of data, one can define the Artin $L$-function $L(V, s)$. Does a similar formula exist for $L(V, s)$?


I would also be interested in a similar formula for the slightly less general case of Hecke $L$-functions.



Explicit formula for Artin L-functions
One proves explicit formulae essentially by integrating the logarithmic derivative of the $L$-function. For simplicity, let $\psi$ be a Schwartz function with $\psi(1) = 1$, and let
$$ \widehat{\psi}(s) = \int\_0^\infty \psi(t) t^{s} \frac{dt}{t} $$
be its Mellin transform. Then one can get an explicit formula by considering
$$ \sum\_{n \geq 1} \Lambda\_V(n) \psi(n) = \frac{-1}{2\pi i} \int\_{(2)} \frac{L'}{L}(V, s) \widehat{\psi}(s) ds. $$
Typically one would shift the line of integration to $\mathrm{Re}(s) = c < 0$, apply the functional equation, and get an expression of the form
$$\begin{align}
\sum\_{n \geq 1} &\big( \Lambda\_V(n) \psi(n) + \Lambda\_\overline{V}(n) \tfrac{\psi(n^{-1})}{n} \big) = \\
&\{ \mathrm{polar\;data} \} - \sum\_\rho \widehat{\psi}(\rho) + \frac{1}{2\pi i} \int\_{(1/2)}\big( \tfrac{\gamma'}{\gamma}(V, s) + \tfrac{\gamma'}{\gamma}(V, 1-s) \big) \widehat{\psi}(s) ds + O(1).
\end{align}$$
Here, I take $\overline{V}$ to be the conjugate representation and assume the functional equation is of the form
$$ Q^{s/2} \gamma(V, s) L(V, s) = \varepsilon(V) Q^{(1-s)/2} \gamma(V, 1-s) L(\overline{V}, 1-s) $$
for collected gamma factors $\gamma(V, s)$ and a root number $\lvert \varepsilon(V) \rvert = 1$. (Brauer's *On Artin's L-series with general group characters* details the general functional equations, and this explicit formula is a in $\S$5.5 of Iwaniec–Kowalski).


But a major challenge in the face of being computationally useful is that we don't know the complete polar data for an arbitrary Artin $L$-function. Assuming Artin's conjecture and taking a nontrivial irreducible representation, there should be no polar contribution and we get a classical explicit formula.


