The simplest of all L-functions is the Riemann zeta function, a function whose analytic properties were first discovered by Riemann in 1859 in his attempts to find a formula for the number of primes smaller than a given number $x$. It gives us a model for how to think about other L-functions.

Let's begin by taking a look at this prototype: the Riemann zeta function homepage.

The first thing we see is that the Riemann zeta function can be defined as a special type of infinite series, called a Dirichlet series:

$\zeta(s):=1+\frac1{2^s}+\frac1{3^s}+\frac1{4^s}+\cdots.$

This is the classical definition of $\zeta(s)$ and holds in the half plane $Re(s)>1$.

All L-functions can be expressed as Dirichlet series. On this website we normalize each Dirichlet series so that it converges absolutely for $Re (s)>1$.

The page then exhibits the functional equation, an equation that relates the values of this L-function on the left and on the right of the line $Re (s)= 1/2$. A key component of the functional equation is that it involves Euler's Gamma function, which is well known as a function that interpolates factorials. Again, this is typical of all L-functions: there is a functional equation involving the product of the L-function and some Gamma functions and relates the values at $s$ and $1-s$. The "Selberg data" is a way of encoding the functional equation; details at the bottom of this page.

The page then goes on by showing the Euler product formula of the L-function. In the case of $\zeta(s)$, this was discovered by Euler and later used by Riemann as a starting point for the investigation of prime numbers. In particular, Riemann showed that the distribution of prime numbers is determined by the location of the zeros of $\zeta(s)$. All L-functions have an Euler product, i.e. they have an expression as an infinite product over primes where the factor for each prime is the inverse of a polynomial in $p^{-s}$.

Next we find the imaginary parts of the first few zeros of the Riemann zeta function. Riemann's Hypothesis is that any non-trivial zero of the zeta function has real part 1/2. Moreover, it's hypothesized that any non-trivial zero of any L-function has real part 1/2. This has been verified for every zero ever found. Consequently on these pages we only display the imaginary parts of the zeros. The first non-trivial zero of the Riemann zeta function is at $1/2+ i 14.1347...$ and was actually computed by Riemann himself. Now it is known that the first 10 trillion zeros of the Riemann zeta function all have real part 1/2. On these pages we have high precision rendering of the first 36 billion zeros; see http://www.lmfdb.org/zeros/zeta/.

Further information found on the Riemann zeta function page is the list of values the function takes at some special points. Again, this is typical of L-functions. The special values of L-functions form a remarkable area of modern research. The Birch and Swinnerton-Dyer Conjecture, mentioned above as one of the Clay Millennium problems, has to do with the special values of L-functions associated with elliptic curves.

Finally the page concludes by displaying the graph of the Hardy Z-function. This function has the same modulus as $\zeta(s)$ on the critical line, so its graph helps to provide a visual picture of the Riemann zeta function. In particular, the Z-function has the same zeros as the L-function and so for this one we can see the first zero at 14.1347...

In order to successfully navigate the L-functions pages you need to know a little bit about how we classify L-functions. As we mentioned, each L-function has a Dirichlet series, a functional equation, and an Euler product. We sort our L-functions first by degree, then by level.

There is a certain amount of compatibility between these expressions. For example, it is believed that the Euler product can always be written as a product over primes of the inverse of a polynomial in $p^{-s}$. In all known cases, at almost all primes the degree of this polynomial is the same as the number of Gamma factors in the functional equation!

We specify a functional equation by providing its Selberg data; this is a 4-tuple which has the shape ( degree, level, Gamma-shifts, sign), or:

$(d,N,(\mu_1,\dots , \mu_{J}:\nu_{1},\dots ,\nu_{K}),\varepsilon);$

Here the degree $d=J+2K$ is a positive integer, the level $N$ is a positive integer, the $\mu$ are complex numbers with non-negative real parts and imaginary parts which sum to 0, and the sign $\epsilon$ is a complex number with absolute value $1$.

For example, the LMFDB page for the Riemann zeta function indicates the Selberg data are (1,1,(0:),1), where the first value 1 is the degree, the second is the level, (0:) indicates the spectral parameters and the last 1 is the sign of the functional equation.