Ramanujan's Dirichlet Lseries 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 .
satisfies the functional equation
(2)

(Hardy 1999, p. 173) and has the Euler product representation
(3)

for (since ) (Apostol 1997, p. 137; Hardy 1999, p. 164).
can be split up into
(4)

where
(5)
 
(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 RiemannSiegel function . The number of zeros in the critical strip from to is given by
(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)
