| L(s) = 1 | − 2.82·2-s + 2.93·3-s + 5.96·4-s + 0.420·5-s − 8.28·6-s − 3.44·7-s − 11.1·8-s + 5.61·9-s − 1.18·10-s − 0.690·11-s + 17.5·12-s − 6.41·13-s + 9.73·14-s + 1.23·15-s + 19.6·16-s + 6.09·17-s − 15.8·18-s + 19-s + 2.51·20-s − 10.1·21-s + 1.94·22-s + 6.89·23-s − 32.8·24-s − 4.82·25-s + 18.1·26-s + 7.66·27-s − 20.5·28-s + ⋯ |
| L(s) = 1 | − 1.99·2-s + 1.69·3-s + 2.98·4-s + 0.188·5-s − 3.38·6-s − 1.30·7-s − 3.95·8-s + 1.87·9-s − 0.375·10-s − 0.208·11-s + 5.05·12-s − 1.78·13-s + 2.60·14-s + 0.319·15-s + 4.91·16-s + 1.47·17-s − 3.73·18-s + 0.229·19-s + 0.561·20-s − 2.20·21-s + 0.415·22-s + 1.43·23-s − 6.70·24-s − 0.964·25-s + 3.55·26-s + 1.47·27-s − 3.88·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 2 | \( 1 + 2.82T + 2T^{2} \) |
| 3 | \( 1 - 2.93T + 3T^{2} \) |
| 5 | \( 1 - 0.420T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 + 0.690T + 11T^{2} \) |
| 13 | \( 1 + 6.41T + 13T^{2} \) |
| 17 | \( 1 - 6.09T + 17T^{2} \) |
| 23 | \( 1 - 6.89T + 23T^{2} \) |
| 29 | \( 1 + 6.37T + 29T^{2} \) |
| 31 | \( 1 + 0.869T + 31T^{2} \) |
| 37 | \( 1 - 6.88T + 37T^{2} \) |
| 41 | \( 1 + 8.97T + 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 0.247T + 53T^{2} \) |
| 59 | \( 1 + 5.51T + 59T^{2} \) |
| 61 | \( 1 + 2.13T + 61T^{2} \) |
| 67 | \( 1 - 0.584T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 7.62T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 0.0438T + 97T^{2} \) |
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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
−8.094542535952996952366917915107, −7.57346879450749137072896672286, −7.20523781820739651563934641070, −6.39581999429943489240981555805, −5.33551499562689191492964378578, −3.48764725507717133101568985262, −3.01564819674028487039770313468, −2.39627541340714024055822905159, −1.45097217193929277786723176546, 0,
1.45097217193929277786723176546, 2.39627541340714024055822905159, 3.01564819674028487039770313468, 3.48764725507717133101568985262, 5.33551499562689191492964378578, 6.39581999429943489240981555805, 7.20523781820739651563934641070, 7.57346879450749137072896672286, 8.094542535952996952366917915107
