| L(s) = 1 | − 2.34·2-s + 3.48·4-s + 1.34·5-s − 7-s − 3.48·8-s − 3.14·10-s − 1.14·11-s + 13-s + 2.34·14-s + 1.19·16-s − 5.83·17-s − 3.34·19-s + 4.68·20-s + 2.68·22-s + 3.17·23-s − 3.19·25-s − 2.34·26-s − 3.48·28-s − 10.4·29-s + 1.63·31-s + 4.17·32-s + 13.6·34-s − 1.34·35-s + 8.51·37-s + 7.83·38-s − 4.68·40-s + 0.292·41-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.74·4-s + 0.600·5-s − 0.377·7-s − 1.23·8-s − 0.994·10-s − 0.345·11-s + 0.277·13-s + 0.626·14-s + 0.299·16-s − 1.41·17-s − 0.766·19-s + 1.04·20-s + 0.572·22-s + 0.662·23-s − 0.639·25-s − 0.459·26-s − 0.659·28-s − 1.94·29-s + 0.293·31-s + 0.738·32-s + 2.34·34-s − 0.226·35-s + 1.40·37-s + 1.27·38-s − 0.740·40-s + 0.0457·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 17 | \( 1 + 5.83T + 17T^{2} \) |
| 19 | \( 1 + 3.34T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 - 1.63T + 31T^{2} \) |
| 37 | \( 1 - 8.51T + 37T^{2} \) |
| 41 | \( 1 - 0.292T + 41T^{2} \) |
| 43 | \( 1 + 8.15T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 0.782T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 6.10T + 67T^{2} \) |
| 71 | \( 1 + 1.53T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 0.882T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 5.73T + 89T^{2} \) |
| 97 | \( 1 + 5.34T + 97T^{2} \) |
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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
−9.502522448235194953977555202779, −9.197734780899711612061834784288, −8.305181942635795144183105679228, −7.42109374514325990843542364952, −6.58321853954104567871249125807, −5.78564899618832235492388940253, −4.29346134336692432418724217176, −2.65660375596362281910892329588, −1.70117854442840435034473668166, 0,
1.70117854442840435034473668166, 2.65660375596362281910892329588, 4.29346134336692432418724217176, 5.78564899618832235492388940253, 6.58321853954104567871249125807, 7.42109374514325990843542364952, 8.305181942635795144183105679228, 9.197734780899711612061834784288, 9.502522448235194953977555202779
