| L(s) = 1 | − 2-s − 0.619·3-s + 4-s + 3.10·5-s + 0.619·6-s + 1.04·7-s − 8-s − 2.61·9-s − 3.10·10-s − 5.96·11-s − 0.619·12-s − 1.66·13-s − 1.04·14-s − 1.91·15-s + 16-s + 4.06·17-s + 2.61·18-s + 6.99·19-s + 3.10·20-s − 0.648·21-s + 5.96·22-s + 23-s + 0.619·24-s + 4.61·25-s + 1.66·26-s + 3.47·27-s + 1.04·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.357·3-s + 0.5·4-s + 1.38·5-s + 0.252·6-s + 0.396·7-s − 0.353·8-s − 0.872·9-s − 0.980·10-s − 1.79·11-s − 0.178·12-s − 0.462·13-s − 0.280·14-s − 0.495·15-s + 0.250·16-s + 0.986·17-s + 0.616·18-s + 1.60·19-s + 0.693·20-s − 0.141·21-s + 1.27·22-s + 0.208·23-s + 0.126·24-s + 0.923·25-s + 0.326·26-s + 0.669·27-s + 0.198·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.243335911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.243335911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 0.619T + 3T^{2} \) |
| 5 | \( 1 - 3.10T + 5T^{2} \) |
| 7 | \( 1 - 1.04T + 7T^{2} \) |
| 11 | \( 1 + 5.96T + 11T^{2} \) |
| 13 | \( 1 + 1.66T + 13T^{2} \) |
| 17 | \( 1 - 4.06T + 17T^{2} \) |
| 19 | \( 1 - 6.99T + 19T^{2} \) |
| 31 | \( 1 - 7.96T + 31T^{2} \) |
| 37 | \( 1 - 4.88T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 - 0.0297T + 43T^{2} \) |
| 47 | \( 1 - 4.23T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 3.77T + 67T^{2} \) |
| 71 | \( 1 + 2.96T + 71T^{2} \) |
| 73 | \( 1 - 5.35T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 3.14T + 83T^{2} \) |
| 89 | \( 1 - 7.94T + 89T^{2} \) |
| 97 | \( 1 - 2.09T + 97T^{2} \) |
| 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
−9.947606979264950413810563789643, −8.888454872862175720647523895327, −7.970511157958071627683756453661, −7.41015139747654478169478055692, −6.14526359184175499256412325351, −5.48901425374711223092130704882, −5.04383134715516224950369877479, −2.96449345844247846481643173165, −2.38169542048888852732582352538, −0.927666849239014986095346697538,
0.927666849239014986095346697538, 2.38169542048888852732582352538, 2.96449345844247846481643173165, 5.04383134715516224950369877479, 5.48901425374711223092130704882, 6.14526359184175499256412325351, 7.41015139747654478169478055692, 7.970511157958071627683756453661, 8.888454872862175720647523895327, 9.947606979264950413810563789643
