| L(s) = 1 | − 2.50·2-s − 2.74·3-s + 4.25·4-s + 3.75·5-s + 6.87·6-s − 1.77·7-s − 5.64·8-s + 4.54·9-s − 9.40·10-s + 4.50·11-s − 11.6·12-s − 2.22·13-s + 4.44·14-s − 10.3·15-s + 5.59·16-s + 2.78·17-s − 11.3·18-s − 6.60·19-s + 15.9·20-s + 4.87·21-s − 11.2·22-s + 5.39·23-s + 15.4·24-s + 9.13·25-s + 5.57·26-s − 4.24·27-s − 7.55·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 1.58·3-s + 2.12·4-s + 1.68·5-s + 2.80·6-s − 0.671·7-s − 1.99·8-s + 1.51·9-s − 2.97·10-s + 1.35·11-s − 3.37·12-s − 0.618·13-s + 1.18·14-s − 2.66·15-s + 1.39·16-s + 0.676·17-s − 2.68·18-s − 1.51·19-s + 3.57·20-s + 1.06·21-s − 2.40·22-s + 1.12·23-s + 3.16·24-s + 1.82·25-s + 1.09·26-s − 0.817·27-s − 1.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 + 2.74T + 3T^{2} \) |
| 5 | \( 1 - 3.75T + 5T^{2} \) |
| 7 | \( 1 + 1.77T + 7T^{2} \) |
| 11 | \( 1 - 4.50T + 11T^{2} \) |
| 13 | \( 1 + 2.22T + 13T^{2} \) |
| 17 | \( 1 - 2.78T + 17T^{2} \) |
| 19 | \( 1 + 6.60T + 19T^{2} \) |
| 23 | \( 1 - 5.39T + 23T^{2} \) |
| 29 | \( 1 - 1.55T + 29T^{2} \) |
| 31 | \( 1 + 4.02T + 31T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 9.73T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 2.95T + 59T^{2} \) |
| 61 | \( 1 - 6.72T + 61T^{2} \) |
| 67 | \( 1 + 3.18T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 6.37T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 6.27T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
| 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
−8.282459472174190397513045406954, −6.95162546962902395687538211426, −6.71371138270106248937206088770, −6.18392877228729763287637560038, −5.53643472903629512095655069292, −4.55562138017775459855923093813, −2.95562752154024858205505555506, −1.79154980794448181307555472662, −1.20540077263469417144753009335, 0,
1.20540077263469417144753009335, 1.79154980794448181307555472662, 2.95562752154024858205505555506, 4.55562138017775459855923093813, 5.53643472903629512095655069292, 6.18392877228729763287637560038, 6.71371138270106248937206088770, 6.95162546962902395687538211426, 8.282459472174190397513045406954
