Dilogarithm Calculator
Free online dilogarithm calculator. Enter your values and get instant results with the formula, step-by-step working, a worked example and FAQs — everything runs locally in your browser on Calcton, no signup required.
中文版：二重对数在线计算器
二重对数 Li₂(z) = −∫₀^z ln(1−t)/t dt 是多对数阶梯的第一级：权重为 2 的「最基本超越积分」。费曼图里单圈顶点的标量积分、体积猜想中的双曲四面体体积、量子场论中 spence 函数，最终都归约到它。它还有个别名 Spence 函数，源于 19 世纪初 Spence 对∫ln²t/(1−t) 的研究。
三条分段路径护航：z ∈ [0, 0.5] 直接级数求和；z ∈ (0.5, 1] 用 Euler 恒等式 Li₂(z) = π²/6 − ln z·ln(1−z) − Li₂(1−z) 折返到收敛区；z 为负时用 Landen 恒等式 Li₂(z) = −Li₂(z/(z−1)) − ½ln²(1−z) 映射回正半轴。三条恒等式互为交叉验证。
参考值
Li₂(0.5) = 0.5822405264650125（= π²/12 − ½ln²2），Li₂(1) = π²/6 = 1.6449340668482264
Li₂(−1) = −π²/12 = −0.8224670334241132（Basel 级数的交错版）
Li₂(1/φ) = π²/10 − ln²φ，Li₂(1/φ²) = π²/15 − ln²φ，其中 φ = (1+√5)/2
How to use the Dilogarithm Calculator
- Fill in the inputs with your own numbers — every field shows what it expects.
- The result updates instantly as you type; no "Calculate" click needed.
- Read the result card for the main figure, the step-by-step formula, and practical notes.
Frequently asked questions
Is the Dilogarithm Calculator free?
Yes — all math calculators on Calcton are free, with no sign-up required. Your inputs are processed in your browser and never sent to a server.
Is there a Chinese version of this tool?
Yes. Use the Chinese version link above — it is the same calculator with the interface in Simplified Chinese.
References
Formulas and benchmarks used by this calculator follow these published sources.
KevinContent LeadCalcton Editorial Team · Last updated 2026-09-27
Kevin maintains Calcton's formula verification and content: every calculator cites the standard it implements, worked examples are recomputed programmatically, and health benchmarks follow WHO and international guidelines.
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