| L(s) = 1 | − 1.73·2-s + 0.999·4-s + 7-s + 1.73·8-s − 3.46·11-s − 5·13-s − 1.73·14-s − 5·16-s − 19-s + 5.99·22-s + 6.92·23-s + 8.66·26-s + 0.999·28-s − 3.46·29-s + 5·31-s + 5.19·32-s + 37-s + 1.73·38-s + 3.46·41-s + 43-s − 3.46·44-s − 11.9·46-s + 3.46·47-s − 6·49-s − 4.99·52-s + 10.3·53-s + 1.73·56-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s + 0.377·7-s + 0.612·8-s − 1.04·11-s − 1.38·13-s − 0.462·14-s − 1.25·16-s − 0.229·19-s + 1.27·22-s + 1.44·23-s + 1.69·26-s + 0.188·28-s − 0.643·29-s + 0.898·31-s + 0.918·32-s + 0.164·37-s + 0.280·38-s + 0.541·41-s + 0.152·43-s − 0.522·44-s − 1.76·46-s + 0.505·47-s − 0.857·49-s − 0.693·52-s + 1.42·53-s + 0.231·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
| 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
−7.73113150341185910820842660006, −7.35820863115049951375983169541, −6.64751101991616912042654855359, −5.42645229258182234815286880670, −4.93224326096937188053609094765, −4.17554336273533188573557838776, −2.81281709134313959588118517057, −2.21764780567849678940369277775, −1.05463291020775159793841319825, 0,
1.05463291020775159793841319825, 2.21764780567849678940369277775, 2.81281709134313959588118517057, 4.17554336273533188573557838776, 4.93224326096937188053609094765, 5.42645229258182234815286880670, 6.64751101991616912042654855359, 7.35820863115049951375983169541, 7.73113150341185910820842660006
