Ramanujan's Dirichlet L-series is defined as
 |
(1)
|
where 
is the tau function. Note that the notation 
is sometimes used instead of 
(Hardy 1999, p. 164).
has properties analogous to the Riemann zeta
function, and is implemented as RamanujanTauL[s].
Ramanujan conjectured that all nontrivial zeros of 
lie on the line ![R[s]=6](/images/equations/TauDirichletSeries/Inline6.svg)
.
satisfies the functional equation
 |
(2)
|
(Hardy 1999, p. 173) and has the Euler product representation
 |
(3)
|
for ![sigma=R[s]>7](/images/equations/TauDirichletSeries/Inline8.svg)
(since 
)
(Apostol 1997, p. 137; Hardy 1999, p. 164).
can be split up into
 |
(4)
|
where
 |  |  |
(5)
|
 |  | ![-1/2iln[(Gamma(6+it))/(Gamma(6-it))]-tln(2pi).](/images/equations/TauDirichletSeries/Inline16.svg) |
(6)
|
The functions 
,
and 
are returned by the Wolfram Language
commands RamanujanTauTheta[t]
and RamanujanTauZ[t],
respectively.
Ramanujan's tau 
-function
is a real function for real 
and is analogous to the Riemann-Siegel
function 
.
The number of zeros in the critical strip from
to 
is given by
![N(t)=(Theta(T)+I{ln[f(6+iT)]})/pi,](/images/equations/TauDirichletSeries/NumberedEquation5.svg) |
(7)
|
where 
is the Ramanujan theta function. Ramanujan conjectured that the nontrivial zeros
of the function are all real.
Ramanujan's 
function is defined by
 |
(8)
|
-Dirichlet Series in the Critical Strip." Math. Comput. 65,
1613-1619, 1996.Spira, R. "Calculation of the Ramanujan Tau-Dirichlet
Series." Math. Comput. 27, 379-385, 1973.Yoshida,
H. "On Calculations of Zeros of L-Functions Related with Ramanujan's Discriminant
Function on the Critical Line." J. Ramanujan Math. Soc. 3, 87-95,
1988.