| L(s) = 1 | + 2-s + 3.38·3-s + 4-s − 2.72·5-s + 3.38·6-s − 2.15·7-s + 8-s + 8.48·9-s − 2.72·10-s + 2.16·11-s + 3.38·12-s − 3.19·13-s − 2.15·14-s − 9.23·15-s + 16-s + 0.0304·17-s + 8.48·18-s + 7.65·19-s − 2.72·20-s − 7.28·21-s + 2.16·22-s + 1.74·23-s + 3.38·24-s + 2.43·25-s − 3.19·26-s + 18.5·27-s − 2.15·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.95·3-s + 0.5·4-s − 1.21·5-s + 1.38·6-s − 0.813·7-s + 0.353·8-s + 2.82·9-s − 0.861·10-s + 0.651·11-s + 0.978·12-s − 0.885·13-s − 0.575·14-s − 2.38·15-s + 0.250·16-s + 0.00738·17-s + 1.99·18-s + 1.75·19-s − 0.609·20-s − 1.59·21-s + 0.460·22-s + 0.364·23-s + 0.691·24-s + 0.486·25-s − 0.626·26-s + 3.57·27-s − 0.406·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.153319051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.153319051\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 - 3.38T + 3T^{2} \) |
| 5 | \( 1 + 2.72T + 5T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 13 | \( 1 + 3.19T + 13T^{2} \) |
| 17 | \( 1 - 0.0304T + 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 23 | \( 1 - 1.74T + 23T^{2} \) |
| 29 | \( 1 - 6.86T + 29T^{2} \) |
| 31 | \( 1 - 4.09T + 31T^{2} \) |
| 37 | \( 1 - 1.14T + 37T^{2} \) |
| 41 | \( 1 + 4.22T + 41T^{2} \) |
| 43 | \( 1 + 4.51T + 43T^{2} \) |
| 47 | \( 1 + 4.11T + 47T^{2} \) |
| 53 | \( 1 - 1.04T + 53T^{2} \) |
| 59 | \( 1 - 9.31T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 4.43T + 71T^{2} \) |
| 73 | \( 1 + 4.68T + 73T^{2} \) |
| 79 | \( 1 + 7.95T + 79T^{2} \) |
| 83 | \( 1 + 2.84T + 83T^{2} \) |
| 89 | \( 1 - 7.87T + 89T^{2} \) |
| 97 | \( 1 - 3.21T + 97T^{2} \) |
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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
−8.276619709537341777255283135404, −7.78312764923715730173874397907, −7.00260719583003036347310482266, −6.70802755690698480282939927550, −5.10877503059162291297062455813, −4.34929647662855149453091664829, −3.60511175319700453087132001727, −3.17207130917301736598683330552, −2.49254880442916464820050555812, −1.14212609400844443651257720147,
1.14212609400844443651257720147, 2.49254880442916464820050555812, 3.17207130917301736598683330552, 3.60511175319700453087132001727, 4.34929647662855149453091664829, 5.10877503059162291297062455813, 6.70802755690698480282939927550, 7.00260719583003036347310482266, 7.78312764923715730173874397907, 8.276619709537341777255283135404
