| L(s) = 1 | + 0.546·2-s − 1.70·4-s + 5-s − 7-s − 2.02·8-s + 0.546·10-s + 11-s + 4.56·13-s − 0.546·14-s + 2.29·16-s − 2.70·17-s − 8.02·19-s − 1.70·20-s + 0.546·22-s + 3.52·23-s + 25-s + 2.49·26-s + 1.70·28-s − 0.133·29-s + 2.33·31-s + 5.29·32-s − 1.47·34-s − 35-s + 6.75·37-s − 4.38·38-s − 2.02·40-s − 10.9·41-s + ⋯ |
| L(s) = 1 | + 0.386·2-s − 0.850·4-s + 0.447·5-s − 0.377·7-s − 0.714·8-s + 0.172·10-s + 0.301·11-s + 1.26·13-s − 0.146·14-s + 0.574·16-s − 0.655·17-s − 1.84·19-s − 0.380·20-s + 0.116·22-s + 0.735·23-s + 0.200·25-s + 0.489·26-s + 0.321·28-s − 0.0247·29-s + 0.420·31-s + 0.936·32-s − 0.253·34-s − 0.169·35-s + 1.11·37-s − 0.711·38-s − 0.319·40-s − 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.754187079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.754187079\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 0.546T + 2T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 + 8.02T + 19T^{2} \) |
| 23 | \( 1 - 3.52T + 23T^{2} \) |
| 29 | \( 1 + 0.133T + 29T^{2} \) |
| 31 | \( 1 - 2.33T + 31T^{2} \) |
| 37 | \( 1 - 6.75T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 9.45T + 43T^{2} \) |
| 47 | \( 1 - 0.649T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 0.959T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 6.41T + 67T^{2} \) |
| 71 | \( 1 - 2.56T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 2.56T + 79T^{2} \) |
| 83 | \( 1 - 7.75T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 1.78T + 97T^{2} \) |
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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
−8.593392867475323857718690607032, −8.171194996616757808792798039689, −6.73661019327337565080515695349, −6.34112437314942744884263299707, −5.59836030287532711611373066117, −4.64181688019237851823049552865, −4.02817925691422543001046621120, −3.21538880955460296704473162342, −2.08813064944516299374867269305, −0.74558737133550264512121241742,
0.74558737133550264512121241742, 2.08813064944516299374867269305, 3.21538880955460296704473162342, 4.02817925691422543001046621120, 4.64181688019237851823049552865, 5.59836030287532711611373066117, 6.34112437314942744884263299707, 6.73661019327337565080515695349, 8.171194996616757808792798039689, 8.593392867475323857718690607032
