Mathematica

Mathematica is a proprietary computer algebra system well-suited to symbolic computation and symbolic manipulation of equations. It also provides facilities for computing numerically to arbitrary precision, but has no built in routines for evaluating multiple zeta values (except for depth 1), nor for general multiple polylogarithms except for the classical polylogarithm \( \Li_n \) and Nielsen polylogarithms \( S_{p,q} \).

Various packages for handling multiple polylogarithms and multiple zeta values have been developed, mainly by the high-energy physics community. For example PolyLogTools developed to compute with the \( \otimes^m \)-symbols of multiple polylogarithms.

Setup
Mathematica is proprietary and not free for general use. It may be available to you via a site-license, or via a number of shared licenses, at your institution. Wolfram Player can run, but cannot edit, pre-existing notebooks.

There is also the possibility of trying Mathematica for 15 days for free here: https://www.wolfram.com/mathematica/trial/

Examples
FindIntegerNullVector is the Mathematica equivalent to the gp/pari command lindep, it finds a linear dependence between input values (to the given precision for numerical values) using PSLQ.

FindIntegerNullVector[{Zeta[2]^2, Zeta[4]}] > {-2, 5}

This indicates that \( 2 \zeta(2)^2 = 5 \zeta(4) \).

Mathematic is better suited to symbolic calcuations. For example, one can implement the stuffle product of MZV's in a very similar way to its recursive mathematical definition as follows. This requires some familiarity with Mathematica pattern matching framework, and is very different to the usual procedural programming which one would see when using pari/gp, say.

app[x_, expr_] := expr /. z[ind___] :> z[x, ind]; st[{}, {w___}] := z[w]; st[ {w___}, {}] := z[w]; st[{a_, w___}, {b_, v___}] := app[a, st[{w}, {b, v}]] + app[b, st[{a, w}, {v}]] + app[a + b, st[{w}, {v}]];

After defining these, one finds that

st[{a, b, c}, {d}]; > z[a, b, c + d] + z[a, b + d, c] + z[a + d, b, c] + z[a, b, c, d] + z[a, b, d, c] + z[a, d, b, c] + z[d, a, b, c]

This means that the stuffle-product \( \zeta(a,b,c) \zeta(d) = \zeta(a, b, c + d) + \zeta(a, b + d, c) + \zeta(a + d, b, c) + \zeta(a, b, c, d) + \zeta(a, b, d, c) + \zeta(a, d, b, c) + \zeta(d, a, b, c) \).