| L(s) = 1 | + 2.67·2-s − 2.48·3-s + 5.15·4-s − 5-s − 6.63·6-s + 7-s + 8.44·8-s + 3.15·9-s − 2.67·10-s − 12.7·12-s − 5.83·13-s + 2.67·14-s + 2.48·15-s + 12.2·16-s − 5.44·17-s + 8.44·18-s + 1.35·19-s − 5.15·20-s − 2.48·21-s − 3.19·23-s − 20.9·24-s + 25-s − 15.5·26-s − 0.387·27-s + 5.15·28-s + 3.61·29-s + 6.63·30-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 1.43·3-s + 2.57·4-s − 0.447·5-s − 2.70·6-s + 0.377·7-s + 2.98·8-s + 1.05·9-s − 0.845·10-s − 3.69·12-s − 1.61·13-s + 0.714·14-s + 0.640·15-s + 3.06·16-s − 1.32·17-s + 1.99·18-s + 0.309·19-s − 1.15·20-s − 0.541·21-s − 0.665·23-s − 4.27·24-s + 0.200·25-s − 3.05·26-s − 0.0746·27-s + 0.974·28-s + 0.670·29-s + 1.21·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 3 | \( 1 + 2.48T + 3T^{2} \) |
| 13 | \( 1 + 5.83T + 13T^{2} \) |
| 17 | \( 1 + 5.44T + 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 + 3.19T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 + 8.54T + 37T^{2} \) |
| 41 | \( 1 - 5.02T + 41T^{2} \) |
| 43 | \( 1 + 5.89T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 0.231T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 1.96T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 7.22T + 89T^{2} \) |
| 97 | \( 1 + 0.836T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.41521569550597947372831559612, −7.00116642646110917256463750055, −6.35011428390453590773251296220, −5.57469812107072089090735828244, −4.88602809142225917595625630127, −4.65297363146386151934892208912, −3.74058465021985711254123288528, −2.66160670152634085717806283879, −1.74103237889578837106585752051, 0,
1.74103237889578837106585752051, 2.66160670152634085717806283879, 3.74058465021985711254123288528, 4.65297363146386151934892208912, 4.88602809142225917595625630127, 5.57469812107072089090735828244, 6.35011428390453590773251296220, 7.00116642646110917256463750055, 7.41521569550597947372831559612
