We establish a short exact sequence about depth-graded motivic double zeta values of even weight relative to $\mu_2$. And we find a basis for the depth-graded motivic double zeta values relative to $\mu_2$ of even weight and a basis for the depth-graded motivic triple zeta values relative to $\mu_2$ of odd weight. As an application of our main results, we prove part of Kaneko and Tasaka's conjectures about the sum odd double zeta values and the classical double zeta values. And we also prove an analogue of part of Kaneko and Tasaka's conjecture in depth three. At last we formulate a conjecture which is related to sum odd multiple zeta values in higher depth.

中文翻译：

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相对于 μ2 的动机倍数 zeta 值

我们建立了一个关于相对于 $\mu_2$ 的偶数权重的深度分级动机双 zeta 值的简短精确序列。并且我们找到了相对于偶数权重 $\mu_2$ 的深度分级动机双 zeta 值的基础，以及相对于奇数权重 $\mu_2$ 的深度分级动机三 zeta 值的基础。作为我们主要结果的应用，我们证明了 Kaneko 和 Tasaka 关于和奇双 zeta 值和经典双 zeta 值的部分猜想。并且我们也在深度三中证明了部分 Kaneko 和 Tasaka 猜想的类比。最后，我们制定了一个猜想，该猜想与更高深度的奇数倍 zeta 值之和有关。