The other part of the
Birch-Swinnerton-Dyer conjecture (which we will
not state here) is a precise formula for the first non-vanishing derivative of
*L*(*E*,*s*) at *s* = 1; it is an example of a
*class number formula*. The earliest examples of such formulas are
related to the failure of unique factorization in number fields and are due
to
Lejeune Dirichlet.
For example, the Dirichlet class number formula for the field
**Q**(√-5) is

where the numerators run over prime numbers^{2}⁄_{2}·^{3}⁄_{2}·^{5}⁄_{5}·^{7}⁄_{6}·^{11}⁄_{12}·^{13}⁄_{14}&hellip =^{&pi}⁄_{√5}

One side of a class number formula is the value of an
*L-function* at an
integer point. In the case of
**Q**(√-5)
the *L*-function is

with (L(Q(√-5),s) := &prod_{p}( 1 - (^{-5}⁄_{p})p^{-s})^{-1}

The right side
consists of three factors. One is a
*regulator*, which is usually
a transcendental number of some sort; in this case it is &pi. The second
is a relatively uninteresting nuisance factor; in this case, it is
actually ^{1}⁄
_{2√5}.
The third factor, 2 in this case, is the *class number*.
In general, the class number should be
related to some interesting arithmetic fact. In this case, it is the size of
the
*ideal class group* of
**Z**[√-5];
the fact that it is different from 1 implies that
**Z**[√-5]
does not have unique factorization, as is also evident from the
famous counterexample

2 · 3 = (1 + √-5) · (1 - √-5).