| L(s) = 1 | + 2.52·3-s − 2.85·5-s − 2.52·7-s + 3.38·9-s − 4.85·13-s − 7.21·15-s + 5.61·17-s − 2.52·19-s − 6.38·21-s + 8.17·23-s + 3.14·25-s + 0.964·27-s + 1.38·29-s + 6.61·31-s + 7.21·35-s − 0.618·37-s − 12.2·39-s + 8.61·41-s − 3.12·43-s − 9.65·45-s + 0.596·47-s − 0.618·49-s + 14.1·51-s + 0.381·53-s − 6.38·57-s + 10.7·59-s − 3.61·61-s + ⋯ |
| L(s) = 1 | + 1.45·3-s − 1.27·5-s − 0.954·7-s + 1.12·9-s − 1.34·13-s − 1.86·15-s + 1.36·17-s − 0.579·19-s − 1.39·21-s + 1.70·23-s + 0.629·25-s + 0.185·27-s + 0.256·29-s + 1.18·31-s + 1.21·35-s − 0.101·37-s − 1.96·39-s + 1.34·41-s − 0.476·43-s − 1.43·45-s + 0.0869·47-s − 0.0882·49-s + 1.98·51-s + 0.0524·53-s − 0.845·57-s + 1.39·59-s − 0.463·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.026576368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.026576368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.52T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 - 5.61T + 17T^{2} \) |
| 19 | \( 1 + 2.52T + 19T^{2} \) |
| 23 | \( 1 - 8.17T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 - 6.61T + 31T^{2} \) |
| 37 | \( 1 + 0.618T + 37T^{2} \) |
| 41 | \( 1 - 8.61T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 - 0.596T + 47T^{2} \) |
| 53 | \( 1 - 0.381T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 3.61T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 4.32T + 73T^{2} \) |
| 79 | \( 1 + 3.49T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 7.23T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.266638090757611544264097677408, −7.927811419621175760566098642460, −7.19866048750955047460657613992, −6.64877364678009841260605754679, −5.32148836969078206203003596151, −4.42494391605597027303607815314, −3.60837390740767859488372968244, −3.05724893123994836814406321052, −2.40332892910761382656649987044, −0.74568143478315124783066145503,
0.74568143478315124783066145503, 2.40332892910761382656649987044, 3.05724893123994836814406321052, 3.60837390740767859488372968244, 4.42494391605597027303607815314, 5.32148836969078206203003596151, 6.64877364678009841260605754679, 7.19866048750955047460657613992, 7.927811419621175760566098642460, 8.266638090757611544264097677408
