L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 3·11-s + 5·13-s − 14-s + 16-s − 7·19-s − 20-s + 3·22-s − 6·23-s + 25-s − 5·26-s + 28-s + 6·29-s − 4·31-s − 32-s − 35-s + 2·37-s + 7·38-s + 40-s + 3·41-s − 43-s − 3·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.904·11-s + 1.38·13-s − 0.267·14-s + 1/4·16-s − 1.60·19-s − 0.223·20-s + 0.639·22-s − 1.25·23-s + 1/5·25-s − 0.980·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.169·35-s + 0.328·37-s + 1.13·38-s + 0.158·40-s + 0.468·41-s − 0.152·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.483842056093934584557644032541, −8.314724688296725481233972144312, −7.50052030654533683484705729992, −6.45582842032306053514721546555, −5.86155685974341315260712076707, −4.64493351951532624617585931009, −3.78445643891231421346868851802, −2.62181111269334866960404615165, −1.52001175376483828990642537367, 0,
1.52001175376483828990642537367, 2.62181111269334866960404615165, 3.78445643891231421346868851802, 4.64493351951532624617585931009, 5.86155685974341315260712076707, 6.45582842032306053514721546555, 7.50052030654533683484705729992, 8.314724688296725481233972144312, 8.483842056093934584557644032541