| L(s) = 1 | − 0.633·2-s + 2.27·3-s − 1.59·4-s + 4.04·5-s − 1.44·6-s − 3.23·7-s + 2.27·8-s + 2.19·9-s − 2.55·10-s − 11-s − 3.64·12-s + 13-s + 2.04·14-s + 9.21·15-s + 1.75·16-s + 1.26·17-s − 1.39·18-s − 3.76·19-s − 6.46·20-s − 7.38·21-s + 0.633·22-s − 4.00·23-s + 5.19·24-s + 11.3·25-s − 0.633·26-s − 1.83·27-s + 5.17·28-s + ⋯ |
| L(s) = 1 | − 0.447·2-s + 1.31·3-s − 0.799·4-s + 1.80·5-s − 0.589·6-s − 1.22·7-s + 0.805·8-s + 0.731·9-s − 0.809·10-s − 0.301·11-s − 1.05·12-s + 0.277·13-s + 0.547·14-s + 2.37·15-s + 0.439·16-s + 0.307·17-s − 0.327·18-s − 0.863·19-s − 1.44·20-s − 1.61·21-s + 0.134·22-s − 0.834·23-s + 1.06·24-s + 2.26·25-s − 0.124·26-s − 0.352·27-s + 0.978·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.259934598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.259934598\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.633T + 2T^{2} \) |
| 3 | \( 1 - 2.27T + 3T^{2} \) |
| 5 | \( 1 - 4.04T + 5T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 23 | \( 1 + 4.00T + 23T^{2} \) |
| 29 | \( 1 - 2.63T + 29T^{2} \) |
| 31 | \( 1 + 3.14T + 31T^{2} \) |
| 37 | \( 1 + 8.50T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 5.18T + 43T^{2} \) |
| 47 | \( 1 + 3.82T + 47T^{2} \) |
| 53 | \( 1 - 0.485T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 3.39T + 71T^{2} \) |
| 73 | \( 1 + 1.58T + 73T^{2} \) |
| 79 | \( 1 - 4.63T + 79T^{2} \) |
| 83 | \( 1 - 9.85T + 83T^{2} \) |
| 89 | \( 1 + 6.51T + 89T^{2} \) |
| 97 | \( 1 - 9.69T + 97T^{2} \) |
| 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
−13.31301314043269552808580546134, −12.75475673412281786777873950641, −10.28138472400765666149597453270, −9.880091714609962268907817656424, −9.040587524671227347252950491810, −8.347837152347321249640225364207, −6.70405565061588825045436935439, −5.43925988716982930693321497323, −3.55396631241579455964387982620, −2.11748065173203137689430082709,
2.11748065173203137689430082709, 3.55396631241579455964387982620, 5.43925988716982930693321497323, 6.70405565061588825045436935439, 8.347837152347321249640225364207, 9.040587524671227347252950491810, 9.880091714609962268907817656424, 10.28138472400765666149597453270, 12.75475673412281786777873950641, 13.31301314043269552808580546134
