What are these special functions?

Question

I'm writing the documentation for my esoteric language Whispers, which specialises in mathematical functions. I've come across the following function which I'm not sure exactly which special function it is, and I can't remember where I found the definition. The code is

def Lc(z):
    return z*ln(2*sin(z*π)) + S(τ*z, 2) / τ

def S(a, b):
    f = lambda k: math.sin(k * a) / k ** b
    return Σ(f, [1, math.inf])

which, in $\LaTeX$ is

$$\text{Lc}(z) = z\ln(2\sin(z\pi)) + \frac{S(\tau z, 2)}\tau \\ S(x, 2) = \sum_{k=1}^\infty \frac{\sin(kx)}{k^2}\\ \text{Lc}(z) = z\ln(2\sin(z\pi)) + \frac{1}\tau \sum_{k=1}^\infty \frac{\sin(\tau kz)}{k^2}$$

Looking through lists such as this or this haven't yielded anything.

$S$ appears to be a related function/extension of the $\text{sinc}$ function, but again, googling around hasn't found anything.

Does anyone recognise either $\text{Lc}(z)$ or $S(a, b)$?

_{I do also have the derivatives and anti-derivatives of both the functions, and the full source code of them can be found here ($\text{Lc}$) and here ($S$)}

Answer 1

For $S(z, 2)$ have a look at the dilogarithm. For other $S(z, b)$, the polylogarithm.

Answer 2

$S(a,\,b)=\sum_{k\ge1}\frac{\sin(ka)}{k^b}$ is a generalized Clausen function, typically denoted $S_b(a)$. In terms of the Clausen function $\operatorname{Cl}_2=S_2$, $\operatorname{Lc}(z)=\tfrac{1}{\tau}\operatorname{Cl}_2(z\tau)-z\operatorname{Cl}_2^\prime(z\tau)$.