I don't know exactly how, but I've read the Riemann Zeta function's nontrivial zero's imply something about an error term for an approximation function thing for the Prime Counting Function. I found it to be really interesting, and I began reading into Dirichlet Characters and their respective L-functions.
But ever since reading about the generalized/grand Riemann Hypothesis, I'm curious to know what is the significance, if any, of the nontrivial zeros of an arbitrary L-function such at the Dirichlet Beta function?
The Riemann zeta function $\zeta(s)$ can be used to prove the prime number theorem (PNT),
$$\pi(x)=\sum_{p\le x}1\sim\frac{x}{\log x}.$$
The proof uses a common trick in analytic number theory: when an arithmetic function is viewed as a "sum of weights," one may reweight the sum in a controlled way so as to analyze a different arithmetic function whose behavior can be converted back into facts about the original arithmetic function. PNT is equivalent to $\psi(x)\sim x$ where $\psi(x)$ is the Chebyshev function defined by
$$\psi(x)=\sum_{p^k\le x}\log p.$$
The so-called explicit formula relates $\psi(x)$ to $\zeta(s)$'s zeros:
$$\psi_0(x)=x-\sum_\rho\frac{x^\rho}{\rho} -\log(2\pi)-\frac{1}{2}\log\left(1-\frac{1}{x^2}\right).$$
(Technical notes: one must sum over the zeros in order of increasing imaginary part. The subscript zero means that $\psi_0(x)$ is the average of $\psi$'s left and right limits at $x$. In other words it counts the last summand with half its usual weight if it is indexed by $x$ itself.)
This formula tells us the error term in using $x$ as an approximation for $\psi(x)$. If the Riemann hypothesis is true, it means we can pull a factor of $\sqrt{x}$ out of the sum and the resulting sum is a kind of Fourier series in the variable $\log x$ whose Fourier frequencies are ${\rm Im}(\rho)$s, but if the RH is not true it means different summands above have different amplifiers $x^{{\rm Re}(\rho)}$ so that summands corresponding to different zeros $\rho$ attain different orders of magnitude of contribution to the counting function $\psi(x)$. This perspective is known as the "music of the primes" - see the book by the same name for more information in a pop-math treatment.
The Dirichlet L-functions $L(s,\chi)$ are the same Dirichlet series but with twisted coefficients, so to speak. If one studies $L(s,\chi)$s for the Dirichlet characters mod $n$ the same way as above, one gets the effective Dirichlet's theorem. DT talks about primes in residue classes of integers mod $n$ (in elementary terms, this means the primes that appear in an infinite arithmetic progression), and is usually presented as stating their exist infinitely many (under an obvious hypothesis below). But the so-called effective version gives a statistical asymptotic statement.
There is an obvious obstruction to the existence of primes congruent to $r\bmod n$: if $r$ is not a unit $\bmod n$ then the only situation where a prime is $r\bmod n$ is when $r$'s least positive integer representative is itself a prime divisor of $n$. This means there is at most one such prime, so we ignore this scenario by assuming our residues are units $\bmod n$.
Effective version of DT states the primes' residues ($p\nmid n$) are equidistributed in $(\Bbb Z/n\Bbb Z)^\times$.
The generalization to Artin L-functions $L(s,\rho)$ (where $\rho$ is an irreducible representation of the Galois group $G_{L/K}$ and $L/K$ is a relative extension of number fields), as you might guess from the parenthetical you just read, requires a lot of technology from algebraic number theory to even build the definition of. I will throw a lot of terms at you that you can google at your leisure.
A field is essentially a number system which has addition, multiplication and all of the usual basic properties in arithmetic that we learn in elementary school. A number field is a field containing the rationals $\Bbb Q$ and which thus becomes a finite-dimensional rational vector space. (Function fields are analogous to number fields, they talk about curves instead of number systems - together number fields and function fields form the class of "global fields" - and pop up in the parallels between arithmetic and geometry.)
Just as $\Bbb Q$ has $\Bbb Z$, number fields $L$ have rings of integers ${\cal O}_L$, the set of algebraic integers (roots of monic polynomials with integer coefficients) in $L$. In these general domains, there is a distinction between prime elements ($p\mid ab\Rightarrow p\mid a$ or $p\mid b$) and irreducible elements (not products of two or more nonunits - they are the atoms of $({\cal O}_K,\times)$ ordered by divisibility). Uniqueness of prime factorizations (when they exist) is trivial, and existence of factorizations into irreducibles is relatively trivial, but in general prime factorizations needn't exist and factorizations into irreducibles needn't be unique (yes, up to order and units). Unique factorization is recovered by looking at ideals of ${\cal O}_L$: all number rings are Dedekind domains, meaning all ideals factor uniquely into prime ideals. So "primes" will simply refer to prime ideals henceforth.
Say $L/K$ is a relative extension of number fields. Given a prime ideal ${\frak p}$ of ${\cal O}_K$, the ideal ${\frak p}{\cal O}_L$ of ${\cal O}_L$ factors into primes ${\frak P}$ of ${\cal O}_L$. Indeed, ${\frak p}{\cal O}_L$ will be unramified (squarefree) $\Leftrightarrow {\frak p}\nmid\Delta_{L/K}$, where $\Delta_{L/K}$ is the relative discriminant, so in particular there are cofinitely many primes of ${\cal O}_K$ that are unramified in ${\cal O}_L$. If $L/K$ is Galois and ${\frak p}$ unramified, then $G_{L/K}$ acts transitively on $\{\frak P\mid p\}$ (the primes ${\frak P}$ of ${\cal O}_L$ appearing in the prime factorization of ${\frak p}{\cal O}_L$). The stabilizer of any given ${\frak P}$ is known as the decomposition group $D_{\frak P\mid p}$.
The residue fields of ${\frak p}$ and ${\frak P}$ are $k={\cal O}_K/{\frak p}$ and $l={\cal O}_L/{\frak P}$. Then $l/k$ is an extension of finite fields. There is an induced isomorphism $D_{\frak P\mid p}\to G_{l/k}$. Pulling back the Frobenius automorphism $x\mapsto x^{|k|}$ of $l/k$ (the canonical generator of the cyclic group $G_{l/k}$) one obtains the Frobenius element $\tau({\frak P})\in D_{\frak P\mid p}\subseteq G_{L/K}$, and thus defines the Artin map
$$\tau:\{\textrm{primes }{\frak P}\textrm{ of }{\cal O}_L\textrm{ such that }{\frak p}={\frak P}\cap{\cal O}_K\textrm{ is unramified}\}\to G_{L/K}.$$
Usually the Artin map is only defined in the context of abelian extensions so as to do class field theory to obtain higher reciprocity laws (generalizing quadratic reciprocity).
Indeed, consider cyclotomic fields $\Bbb Q(\zeta)/\Bbb Q$ where $\zeta$ is a primitive $n$th root of unity. Then the Galois group $G$ may be canonically identified with $(\Bbb Z/n\Bbb Z)^\times$ (not merely isomorphic with, mind you). In this way, the Artin map assigns to a prime ${\frak P}\mid p\Bbb Z[\zeta]$ ($p\in \Bbb Z$) the corresponding residue class of $p$ in the Galois group $(\Bbb Z/n\Bbb Z)^\times$. The condition that we only look at unramified primes is equivalent to our previous condition of only looking at primes which are units mod $n$! In this way, our previous effective Dirichlet theorem generalizes to the effective Chebotarev density theorem.
In order to make sense of the effective version of these theorems, we need a way to describe the "density" of sets $T$ of primes (within the total collection of all unramified primes)...
Natural density. Ideals $\frak I$ of ${\cal O}_L$ have a norm $N({\frak J})\in\Bbb N$ defined for them, generalizing the "product of conjugates" one uses to simplify denominators with radicals. Then $$\delta_{{\rm nat}}(T)=\lim_{n\to\infty}\frac{\{{\frak P}\in T:N({\frak P})\le n\}}{\{{\frak P}:N({\frak P})\le n\}}. $$
Polar density. Set $\zeta_{L,T}(s)=\prod\limits_{{\frak P}\in T}(1-N({\frak P})^{-s})^{-1}.\,$ If $\zeta_{L,T}(s)^n$ has a meromorphic continuation to a neighborhood of $s=1$ with pole of order $m$, then $\delta_{\rm pol}(T)=m/n$.
Dirichlet density. This is defined by the limiting ratio of restricted Dirichlet series: $$\delta_{{\rm Dir}}(T)=\lim_{s\to1^+}\left(\sum_{{\frak P}\in T} \frac{1}{N({\frak P})^s}\right)/\left(\sum_{\frak P} \frac{1}{N({\frak P})^s}\right).$$
Formally, there is a different version of the density theorem for different densities. The natural density is the hardest to prove anything about, one instead uses the Dirichlet density or polar density. Indeed, the Dirichlet density extends the other two densities - meaning, if (say) $\delta_{\rm nat}(T)$ exists, then so does $\delta_{\rm Dir}(T)$ and $\delta_{\rm Dir}(T)=\delta_{\rm nat}(T)$.
Effective Chebotarev Density Theorem. The densities of the fibers of the Artin map induce the uniform distribution on the Galois group. That is, the set of primes $\frak P$ of ${\cal O}_L$ whose Frobenius element $\tau({\frak P})$ is a given element $g\in G_{L/K}$ has density $1/|G_{L/K}|$ for any choice of $g$.
This is almost always stated in terms of primes of ${\cal O}_K$ and conjugacy classes in $G_{L/K}$, but I'm pretty sure this version is equivalent and certainly more elegant (I don't recall if I read it somewhere or just started believing it). Given a prime $\frak p$ of ${\cal O}_K$, define $\tau({\frak p})$ to be the conjugacy class of $G_{L/K}$ traced out by $\tau({\frak P})$ as $\frak P$ varies over prime divisors of ${\frak p}{\cal O}_L$. The usual form says that the density of ${\frak p}$ in ${\cal O}_K$ with $\tau({\frak})$ a given conjugacy class $C$ is given by $|G|/|G_{L/K}|$.
Now, so far I haven't actually mentioned what the $L$-functions zeros are, or even what the $L$-functions in question are! Here, we speak of Artin $L$-functions - one may think of them as the (in some sense) "factors" of the Dedekind zeta function $\zeta_K(s)$. If we have a Galois representation $\rho:G_{L/K}\to {\rm GL}(V)$ (for us, irreducible) then $\rho(g)$'s conjugacy class depends only on $g$s conjugacy class in $G_{L/K}$, so the following makes sense:
$$L(s,\rho)=\prod_{{\frak p}\textrm{ unr}}\det\left(1-\frac{1}{N({\frak p})^s}\rho(\tau({\frak p}))\right)^{-1}.$$
Since irreducible representations are uniquely determined by their character $\chi$, sometimes this is denoted instead by $L(s,\chi)$. Other times we attach even more notation to the expression $L(s,\chi)$, for instance the extension $L/K$ or the carrier space $V$ of the representation.
One uses Dirichlet $L$-functions to prove effective Dirichlet's theorem, and similarly one uses Artin $L$-functions to prove the effective Chebotarev density theorem. Just as with the Riemann zeta function when the zeros were used to describe the reweighted counting function $\psi(x)$, we also have a "twisted" version of $\psi$ defined by
$$\psi(x,\rho)=\sum_{\substack{{\frak p}\textrm{ unram} \\ N({\frak p})^m\le x}}{\rm tr}\,\rho(\tau({\frak p}))\log N({\frak p}). $$
There is an explicit formula for $\psi(x,\rho)$ in terms of $L(s,\rho)$'s zeros. See Theorem $3.4.9$ of this thesis to see it written out - it's bigger and involves more terms I haven't explained. (Also since the letter $\rho$ is used for zeros, its notation refers to characters $\chi$ instead of representations $\rho$.)
To read more about this, obtain almost any relatively advanced book on algebraic number theory, or google "Artin $L$-functions," "Chebotarev density," or both for many pdfs on the web. Surely there are (fuzzily) analogous statements that can be said about zeros of other types of zeta functions too, but I don't know as much about other types of zeta functions.