| L(s) = 1 | + 17.3·5-s − 2.37·7-s − 52.2·11-s + 13.6·13-s − 82.9·17-s − 126.·19-s − 54.3·23-s + 174.·25-s − 212.·29-s − 224.·31-s − 41.1·35-s − 32.2·37-s + 500.·41-s − 12.8·43-s + 209.·47-s − 337.·49-s + 371.·53-s − 904.·55-s + 5.81·59-s − 604.·61-s + 237.·65-s + 754.·67-s + 43.4·71-s + 671.·73-s + 123.·77-s − 649.·79-s + 134.·83-s + ⋯ |
| L(s) = 1 | + 1.54·5-s − 0.128·7-s − 1.43·11-s + 0.292·13-s − 1.18·17-s − 1.53·19-s − 0.492·23-s + 1.39·25-s − 1.36·29-s − 1.29·31-s − 0.198·35-s − 0.143·37-s + 1.90·41-s − 0.0454·43-s + 0.650·47-s − 0.983·49-s + 0.962·53-s − 2.21·55-s + 0.0128·59-s − 1.26·61-s + 0.452·65-s + 1.37·67-s + 0.0726·71-s + 1.07·73-s + 0.183·77-s − 0.924·79-s + 0.177·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 17.3T + 125T^{2} \) |
| 7 | \( 1 + 2.37T + 343T^{2} \) |
| 11 | \( 1 + 52.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 13.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 82.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 126.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 54.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 212.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 224.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 32.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 500.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 12.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 209.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 371.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 5.81T + 2.05e5T^{2} \) |
| 61 | \( 1 + 604.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 754.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 43.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 671.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 649.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 134.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 206.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.45e3T + 9.12e5T^{2} \) |
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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
−9.666151242056943768818209944779, −9.037496011950889746098695082261, −8.053624878528014428711411581768, −6.90050237710058172144609500092, −5.97236565732953688973506798805, −5.37173042256461412627476492712, −4.14547165233202423352919932794, −2.53074320587176627726990986953, −1.91171122375780888228158589491, 0,
1.91171122375780888228158589491, 2.53074320587176627726990986953, 4.14547165233202423352919932794, 5.37173042256461412627476492712, 5.97236565732953688973506798805, 6.90050237710058172144609500092, 8.053624878528014428711411581768, 9.037496011950889746098695082261, 9.666151242056943768818209944779
