Particular values of the Riemann zeta function

In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted and is named after the mathematician Bernhard Riemann. When the argument is a real number greater than one, the zeta function satisfies the equation
It can therefore provide the sum of various convergent infinite series, such as Explicit or numerically efficient formulae exist for at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments.
The same equation in above also holds when is a complex number whose real part is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the complex plane by analytic continuation, except for a simple pole at. The complex derivative exists in this more general region, making the zeta function a meromorphic function. The above equation no longer applies for these extended values of, for which the corresponding summation would diverge. For example, the full zeta function exists at, but the corresponding series would be, whose partial sums would grow indefinitely large.
The zeta function values listed below include function values at the negative even numbers, for which and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane. The successful characterisation of its non-trivial zeros in the wider plane is important in number theory, because of the Riemann hypothesis.

The Riemann zeta function at 0 and 1

At zero, one has
At 1 there is a pole, so is not finite but the left and right limits are:
Since it is a pole of first order, it has a complex residue

Positive integers

Even positive integers

For the even positive integers, one has the relationship to the Bernoulli numbers :
The computation of is known as the Basel problem. The value of is related to the Stefan–Boltzmann law and Wien approximation in physics. The first few values are given by:
Taking the limit, one obtains.

Value	Decimal expansion	Source

The relationship between zeta at the positive even integers and powers of pi may be written as
where and are coprime positive integers for all. These are given by the integer sequences and, respectively, in OEIS. Some of these values are reproduced below:

n	a_n	b_n
1	6	1
2	90	1
3	945	1
4	9450	1
5	93555	1
6	638512875	691
7	18243225	2
8	325641566250	3617
9	38979295480125	43867
10	1531329465290625	174611
11	13447856940643125	155366
12	201919571963756521875	236364091
13	11094481976030578125	1315862
14	564653660170076273671875	6785560294
15	5660878804669082674070015625	6892673020804
16	62490220571022341207266406250	7709321041217
17	12130454581433748587292890625	151628697551

If we let be the coefficient of as above,
then we find recursively,
This recurrence relation may be derived from that for the Bernoulli numbers.
Also, there is another recurrence:
which can be proved, using that
The values of the zeta function at non-negative even integers have the generating function:
Since
The formula also shows that for,

Odd positive integers

The sum of the harmonic series is infinite.
The value is also known as Apéry's constant and has a role in the electron's gyromagnetic ratio.
The value also appears in Planck's law.
These and additional values are:

Value	Decimal expansion	Source
[Apéry's constant\|]

It is known that is irrational and that infinitely many of the numbers, are irrational. There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of or is irrational.
The positive odd integers of the zeta function appear in physics, specifically correlation functions of antiferromagnetic XXX spin chain.
Most of the identities following below are provided by Simon Plouffe. They are notable in that they converge quite rapidly, giving almost three digits of precision per iteration, and are thus useful for high-precision calculations.
Plouffe stated the following identities without proof. Proofs were later given by other authors.

''ζ''(7)

Note that the sum is in the form of a Lambert series.

''ζ''(2''n'' + 1)

By defining the quantities
a series of relationships can be given in the form
where and are positive integers. Plouffe gives a table of values:

n	a_n	b_n	c_n	d_n
3	180	7	360	0
5	1470	5	3024	84
7	56700	19	113400	0
9	18523890	625	37122624	74844
11	425675250	1453	851350500	0
13	257432175	89	514926720	62370
15	390769879500	13687	781539759000	0
17	1904417007743250	6758333	3808863131673600	29116187100
19	21438612514068750	7708537	42877225028137500	0
21	1881063815762259253125	68529640373	3762129424572110592000	1793047592085750

These integer constants may be expressed as sums over Bernoulli numbers, as given in below.
A fast algorithm for the calculation of Riemann's zeta function for any integer argument is given by E. A. Karatsuba.

Negative integers

In general, for negative integers, one has
The so-called "trivial zeros" occur at the negative even integers:

The first few values for negative odd integers are
This means can be used as the definition of Bernoulli numbers.
However, just like the Bernoulli numbers, these do not stay small for increasingly negative odd values. For details on, see 1 + 2 + 3 + 4 + ⋯.

Derivatives

The derivative of the zeta function at the negative even integers is given by
The first few values of which are
One also has
where is the Glaisher–Kinkelin constant. The first of these identities implies that the regularized product of the reciprocals of the positive integers is, thus the amusing "equation".
From the logarithmic derivative of the functional equation,

Value	Decimal expansion	Source

Series involving ''ζ''(''n'')

The following sums can be derived from the generating function:
where is the digamma function.
Series related to the Euler–Mascheroni constant are
and using the principal value
which of course affects only the value at 1, these formulae can be stated as
and show that they depend on the principal value of.

Nontrivial zeros

Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". The Riemann hypothesis states that the real part of every nontrivial zero must be. In other words, all known nontrivial zeros of the Riemann zeta are of the form where is a real number. The following table contains the decimal expansion of for the first few nontrivial zeros:

Decimal expansion of Im	Source

Andrew Odlyzko computed the first 2 million nontrivial zeros accurate to within, and the first 100 zeros accurate within 1000 decimal places. See their website for the tables and bibliographies.
A table of about 103 billion zeros with high precision is available for interactive access and download in a compressed format via LMFDB.

Ratios

Although evaluating particular values of the zeta function is difficult, often certain ratios can be found by inserting particular values of the gamma function into the functional equation
We have simple relations for half-integer arguments
Other examples follow for more complicated evaluations and relations of the gamma function. For example a consequence of the relation
is the zeta ratio relation
where AGM is the arithmetic–geometric mean. In a similar vein, it is possible to form radical relations, such as from
the analogous zeta relation is