and 3-powers, in which case standard relations seem to be often incomplete whereas Deligne shows that their bound should be sharp by a variant of Grothedieck’s period conjecture. This suggests that in general there should be other linear relations among MPVs besides the standard relations, some of which are written down in this paper explicitly with good numerical verification. We also provide a few conjectures which are supported by our computational evidence.

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Introduction
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In recent years, there is a revival of interest in multi-valued classical polylogarithms (polylogs) and their generalizations. For any positive integers $s_1,\dots, s_\ell$, Goncharov [@Gicm] defines the multiple polylogs of complex variables as follows: $$\label{equ:polylog}
Li_{s_1,\dots, s_\ell}(x_1,\dots,x_\ell)=\sum_{k_1>\dots>k_\ell>0}
\frac{x_1^{k_1}\cdots x_\ell^{k_\ell}}{ k_1^{s_1}\cdots k_\ell^{s_\ell}}.$$ Conventionally one calls $\ell$ the [*depth*]{} (or [*length*]{}) and $s_1+\dots+s_\ell$ the [*weight*]{}. When the depth $\ell=1$ the function is nothing but the classical polylog. When the weight is also 1 we get the MacLaurin series of $-\log(1-x)$. Another useful expression of the multiple polylogs is given by the following iterated integral: $$\label{equ:iteratedForm}
Li_{s_1,\dots,s_\ell}(x_1,\dots,x_\ell)=(-1)^\ell
\int_{0}^1 \left(\frac{dt}{t}\right)^{\circ(s_1-1)}\circ\frac{d t}{t-a_1}
\circ\cdots \circ\left(\frac{dt}{t}\right)^{\circ(s_\ell-1)}\circ\frac{d t}{t-a_\ell}$$ where $a_i=1/(x_1\dots x_i)$ for $1\le i\le \ell$. Here, we define the iterated integrals recursively by $\int_a^b f(t)\circ w(t) =\int_a^b (\int_a^x f(t)) w(x)$ for any 1-form 