| L(s) = 1 | + 0.414·3-s + 5-s + 0.414·7-s − 2.82·9-s + 2·13-s + 0.414·15-s − 5.65·17-s + 0.828·19-s + 0.171·21-s + 7.65·23-s + 25-s − 2.41·27-s + 2·29-s − 5.65·31-s + 0.414·35-s − 4·37-s + 0.828·39-s + 7.48·41-s − 6.41·43-s − 2.82·45-s − 10.4·47-s − 6.82·49-s − 2.34·51-s + 3.65·53-s + 0.343·57-s − 11.3·59-s + 61-s + ⋯ |
| L(s) = 1 | + 0.239·3-s + 0.447·5-s + 0.156·7-s − 0.942·9-s + 0.554·13-s + 0.106·15-s − 1.37·17-s + 0.190·19-s + 0.0374·21-s + 1.59·23-s + 0.200·25-s − 0.464·27-s + 0.371·29-s − 1.01·31-s + 0.0700·35-s − 0.657·37-s + 0.132·39-s + 1.16·41-s − 0.978·43-s − 0.421·45-s − 1.51·47-s − 0.975·49-s − 0.328·51-s + 0.502·53-s + 0.0454·57-s − 1.47·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 7 | \( 1 - 0.414T + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 - 0.828T + 19T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 7.48T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 3.58T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 2.48T + 79T^{2} \) |
| 83 | \( 1 + 9.31T + 83T^{2} \) |
| 89 | \( 1 + 8.65T + 89T^{2} \) |
| 97 | \( 1 - 5.31T + 97T^{2} \) |
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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
−7.29730307113811949948465865313, −6.58866008155836280309450883840, −6.07267795248099846897425197871, −5.19215533668999173471710283176, −4.74539390599279573730079697576, −3.65809382980077643966243739431, −3.00553733459205509686764162969, −2.22281723975367211310787491714, −1.32895528378198871942720540511, 0,
1.32895528378198871942720540511, 2.22281723975367211310787491714, 3.00553733459205509686764162969, 3.65809382980077643966243739431, 4.74539390599279573730079697576, 5.19215533668999173471710283176, 6.07267795248099846897425197871, 6.58866008155836280309450883840, 7.29730307113811949948465865313
