Dirichlet Beta Calculator
Free online dirichlet beta 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.
β(s) = 1 − 1/3ˢ + 1/5ˢ − 1/7ˢ + … 是模 4 狄利克雷 L 函数 L(s, χ₄)，s = 1 时就是莱布尼茨求 π 的级数。
交错奇分母的 zeta 亲戚
β(2) = 0.9159655941772190…（OEIS A006752）——卡塔兰常数至今没有发现闭式
把 ζ(s) 的偶数项抽走再交错变号，剩下的就是 β(s)：它筛出 4k+1 与 4k+3 型素数的信息，是 L(s, χ₄)。本页用欧拉变换把收敛速度从每项 O(1/nˢ) 提到 O(1/4ⁿ)，48 项即可达 1e-13——β(1) 直接复现 π/4 的 13 位有效数字。几何上 β(2) 出现在格点随机游走常数与扭结能量里。
How to use the Dirichlet Beta 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 Dirichlet Beta 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.