| L(s) = 1 | − 0.372·2-s − 1.86·4-s + 1.42·5-s + 1.43·8-s − 0.529·10-s + 3.76·11-s + 1.86·13-s + 3.18·16-s − 7.53·17-s − 1.67·19-s − 2.64·20-s − 1.40·22-s − 0.511·23-s − 2.97·25-s − 0.693·26-s − 8.72·29-s − 2.93·31-s − 4.06·32-s + 2.80·34-s + 4.33·37-s + 0.624·38-s + 2.04·40-s + 6.84·41-s − 4.53·43-s − 7.00·44-s + 0.190·46-s − 7.43·47-s + ⋯ |
| L(s) = 1 | − 0.263·2-s − 0.930·4-s + 0.635·5-s + 0.508·8-s − 0.167·10-s + 1.13·11-s + 0.516·13-s + 0.796·16-s − 1.82·17-s − 0.384·19-s − 0.591·20-s − 0.299·22-s − 0.106·23-s − 0.595·25-s − 0.136·26-s − 1.61·29-s − 0.526·31-s − 0.718·32-s + 0.481·34-s + 0.712·37-s + 0.101·38-s + 0.323·40-s + 1.06·41-s − 0.691·43-s − 1.05·44-s + 0.0281·46-s − 1.08·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.372T + 2T^{2} \) |
| 5 | \( 1 - 1.42T + 5T^{2} \) |
| 11 | \( 1 - 3.76T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 + 7.53T + 17T^{2} \) |
| 19 | \( 1 + 1.67T + 19T^{2} \) |
| 23 | \( 1 + 0.511T + 23T^{2} \) |
| 29 | \( 1 + 8.72T + 29T^{2} \) |
| 31 | \( 1 + 2.93T + 31T^{2} \) |
| 37 | \( 1 - 4.33T + 37T^{2} \) |
| 41 | \( 1 - 6.84T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 + 7.43T + 47T^{2} \) |
| 53 | \( 1 + 0.832T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 6.51T + 67T^{2} \) |
| 71 | \( 1 + 4.11T + 71T^{2} \) |
| 73 | \( 1 - 5.69T + 73T^{2} \) |
| 79 | \( 1 - 0.264T + 79T^{2} \) |
| 83 | \( 1 - 7.90T + 83T^{2} \) |
| 89 | \( 1 + 0.796T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.339733147028185277651165611324, −7.40788257238664559736405018327, −6.51129953910438150934519013402, −5.95411163239373925081626069351, −5.05557301060599780296141997990, −4.15354854845866238481336312038, −3.72140152173220701665678628507, −2.23684433038896207050193428539, −1.40664528018848430030965982015, 0,
1.40664528018848430030965982015, 2.23684433038896207050193428539, 3.72140152173220701665678628507, 4.15354854845866238481336312038, 5.05557301060599780296141997990, 5.95411163239373925081626069351, 6.51129953910438150934519013402, 7.40788257238664559736405018327, 8.339733147028185277651165611324
