Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a\_1, a\_2, a\_3, \cdots$ such that 


* the corresponding L-function
$$L\_{\{a\_n\}}(s):=\sum\_{n=1}^{\infty}\frac{a\_n}{n^s}$$
converges well for $\text{Re}(s) > M$ for some $M\in \mathbb{R}$
* $L\_{\{a\_n\}}(s)$ has analytic continuation to a meromorphic function on the whole complex plane
* $L\_{\{a\_n\}}(n)=0$ for all negative integers $n$
* not all of $a\_n$ are zero?


**Added** : It was suggested in the answers that I should have used the term "Dirichlet series of integer sequences" instead of "L-function" as it lacks Euler product. I apologize for the confusion :)



Are L-functions uniquely determined by their values at negative integers?

> 
> Are L-functions uniquely determined by their values at negative integers?
> 
> 
> 


No. The rescaled Riemann zeta function
$$
\zeta(2s) = \sum\_{m=1}^\infty \frac{1}{m^{2s}} =
\sum\_{n=1}^\infty \frac{a\_n}{n^s},
$$
corresponding to the coefficient sequence
$$
a\_n = \begin{cases}
1 & \textrm{if $n$ is a square}, \\
0 & \textrm{otherwise},
\end{cases}
$$
is an example of an $L$-function that has a meromorphic continuation to all of $\mathbb{C}$ and vanishes at the negative integers.


Note that the formulation of your question seems to mix up the notion of an $L$-function with the more general notion of a Dirichlet series.


There are also some interesting Dirichlet series that are not $L$-functions but still satisfy the properties you are asking about (meromorphic continuation and zeros at the negative integers). One such function is the so-called Witten zeta function of the group $SU(3)$, as I proved in "[On the number of $n$-dimensional representations of $\operatorname{SU}(3)$, the Bernoulli numbers, and the Witten zeta function](https://www.math.ucdavis.edu/~romik/data/uploads/papers/su3reps-final.pdf)" (see theorem 1.3 on page 5). The coefficient sequence for that function is
\begin{align}
a\_n &= \#\{ j,k\ge 1 : n = jk(j+k) \}
\\ &= 
\textrm{the number of inequivalent irreducible}
\\ & \quad \textrm{ representations of $SU(3)$ of dimension $n/2$.}
\end{align}


