| L(s) = 1 | + 19.8·3-s + 160.·7-s + 149.·9-s − 404.·11-s − 249.·13-s − 1.25e3·17-s − 905.·19-s + 3.18e3·21-s + 2.15e3·23-s − 1.85e3·27-s − 2.79e3·29-s + 3.31e3·31-s − 8.00e3·33-s − 1.31e4·37-s − 4.94e3·39-s − 5.07e3·41-s − 1.18e4·43-s − 1.61e3·47-s + 9.09e3·49-s − 2.47e4·51-s − 3.35e4·53-s − 1.79e4·57-s + 2.21e4·59-s − 4.32e4·61-s + 2.39e4·63-s + 4.40e4·67-s + 4.25e4·69-s + ⋯ |
| L(s) = 1 | + 1.27·3-s + 1.24·7-s + 0.613·9-s − 1.00·11-s − 0.409·13-s − 1.05·17-s − 0.575·19-s + 1.57·21-s + 0.847·23-s − 0.490·27-s − 0.616·29-s + 0.619·31-s − 1.27·33-s − 1.58·37-s − 0.520·39-s − 0.471·41-s − 0.976·43-s − 0.106·47-s + 0.541·49-s − 1.33·51-s − 1.64·53-s − 0.731·57-s + 0.828·59-s − 1.48·61-s + 0.761·63-s + 1.19·67-s + 1.07·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 19.8T + 243T^{2} \) |
| 7 | \( 1 - 160.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 404.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 249.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.25e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 905.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.31e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.31e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.07e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.18e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.61e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.35e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.32e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.72e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.57e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.21e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.57e5T + 8.58e9T^{2} \) |
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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
−8.824358729273971593204415470553, −8.307791316976726273780183200971, −7.66845197490403660130363228789, −6.72687245059835227268533591810, −5.24905896427456969467728693620, −4.60052651197799418602174093946, −3.38419693014268968981035546602, −2.38681217805345838277867776726, −1.70058956699222226167089484582, 0,
1.70058956699222226167089484582, 2.38681217805345838277867776726, 3.38419693014268968981035546602, 4.60052651197799418602174093946, 5.24905896427456969467728693620, 6.72687245059835227268533591810, 7.66845197490403660130363228789, 8.307791316976726273780183200971, 8.824358729273971593204415470553
