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" (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}

Your restriction that all $a_n$ are integers is too restrictive to define $L$-functions in general (even most Dirichlet $L$-functions are not like that), and you left out the Euler product. Anyway, examples with the restrictions you imposed can be found among the zeta-functions of number fields.

For a number field $F$, its zeta-function $\zeta_F(s)$ has order of vanishing $r_2(F)$ (the number of conjugate pairs of complex embeddings of $F$) at negative odd integers and $r_1(F) + r_2(F)$ (here $r_1(F)$ is the number of real embeddings of $F$) at negative even integers. Always $r_1(F) + r_2(F)$ is positive, and $r_2(F) > 0$ exactly when $F$ is not totally real, so $\zeta_F(s)$ vanishes at all negative integers when $F$ is not totally real. For example, the zeta-function of $\mathbf Q(i)$, or more generally any cyclotomic field other than $\mathbf Q$ itself, vanishes at all negative integers. The zeta-function of a number field is not identically 0 as a function since it tends to 1 (its constant term) as ${\rm Re}(s) \rightarrow \infty$ or since $a_p = [F:\mathbf Q]$ when $p$ is a prime splitting completely in $F$ (there are infinitely many such $p$).

The zeta-function of $\mathbf Q(i)$ equals the product $\zeta(s)L(s,\chi_4)$ where $\chi_4$ is the nontrivial character mod $4$, where $\zeta(s)$ vanishes at negative even integers but not negative odd integers and $L(s,\chi_4)$ vanishes at negative odd integers but not negative even integers. So you might feel that something like $\zeta_{\mathbf Q(i)}(s)$ is not a good example since it is a product where each piece ($\zeta(s)$ or $L(s,\chi_4)$) is not an example of the kind you seek. You could use instead $L$-functions of elliptic curves (over $\mathbf Q$, say) all of which satisfy the conditions you impose and most (the ones for non-CM elliptic curves) are not expected to break up into two parts which vanish only on negative even or negative odd integers. Or use the Artin $L$-function of the $2$-dimensional irreducible representation of ${\rm Gal}(F/\mathbf Q) \cong S_3$ where $F$ is the splitting field over $\mathbf Q$ of $x^3-2$: it satisfies all of your conditions.

The answer by David Loeffler here gives a formula for the order of vanishing at negative integers of Hecke $L$-functions over a number field $F$. From the formula you see that the order of vanishing is positive at all negative integers if $F$ is not totally real. As he writes, "the orders of vanishing for negative integers $s$ are completely determined by the Gamma-factors in the functional equation" so when you have a functional equation that involves $\Gamma(s)$, which has poles at all negative integers, this forces there to be zeros of the $L$-function at negative integers since the product of the (dual) $L$-function and $\Gamma$-factor is nonzero at positive integers to the right of the critical strip and then you use the functional equation to relate that to values at negative integers,

Let $F(s) =\sum_{n=1}^\infty a_n n^{-s}= \sum_{j=1}^J c_j F_j(s)$ be a linear combination of L-functions $F_j$ of degree $\le d$, that is $(s-1)^r F_j(s)$ entire and its functional equation contains at most $d$ gamma factors $\Gamma(s/2+a)$.

Then for $m > d$ or for $m=d,|x| < C$, $\Gamma(ms)F(s)x^{-s}$ decays fast enough to apply the residue theorem $$\begin{eqnarray}f_m(x)&=&\frac{1}{2i\pi} \int_{2-i\infty}^{2+i\infty} F(s) \Gamma(ms) x^{-s}ds \\ &=& \sum Res(F(s) \Gamma(ms) x^{-s}) \\ &=& \sum Res_{s=1}(F(s) \Gamma(ms)x^{-s}) + \sum_{k=0}^\infty \frac{(-1)^k}{k!m} F(-k/m) x^{k/m}\end{eqnarray}$$ (for $m = d$ it is valid only for $|x| < C$ but the exponential decay of $\Gamma(ms)F(s)$ on $\Im(s)=2$ implies $f_m$ is analytic so it is determined by $x \in (0,C)$)

And hence $$F(s) = \frac{1}{\Gamma(ms)} \int_0^\infty f_m(x) x^{s-1}dx$$

is fully determined by its principal part at $1$ and its values at $s=-k/m$.

In other words your claim holds only for L-functions of degree $d=1$.