Hurwitz Zeta Calculator
Free online hurwitz zeta 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.
Hurwitz zeta：ζ(s, a) = Σ 1/(k+a)^s（k 从 0 起），是黎曼 ζ 的"平移版"，也是多伽马与 L 函数的母体。
从黎曼 ζ 出发的第一步推广
ζ(s, a) = a^(−s) + ζ(s, a+1)；ζ(s, 1/2) = (2^s − 1)ζ(s)（抽走偶数项得奇数项和）
Hurwitz 1882 年把黎曼级数的起点从 1 挪到任意 a：a = 1 还原黎曼 ζ，a = 1/2 得奇数项和，a = 1/q 配上相位因子 e^(−2πiqa/q) 就线性组合出狄利克雷 L 函数——β(s) 正是 χ₄ 的那个组合。解析延拓后 a 只能取正实数（整数 a 之外再延拓会撞上多值性），物理里的 Bose–Einstein 积分、Casimir 效应与 zeta 正则化全靠它。
How to use the Hurwitz Zeta 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 Hurwitz Zeta 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 2026-09-26
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.
Related calculators
More in this category: Math calculators
About math calculators
Free online math calculators for algebra, statistics, number theory and everyday arithmetic — from quadratic equations and matrices to percentages, fractions and prime factorization, each with step-by-step formulas.