| L(s) = 1 | − 3.04·5-s − 0.418·7-s − 3·9-s + 5.27·11-s − 2.62·13-s + 3.27·17-s − 19-s − 3.46·23-s + 4.27·25-s − 2.62·29-s + 6.09·31-s + 1.27·35-s − 9.55·37-s − 4.54·41-s + 2.72·43-s + 9.13·45-s − 0.418·47-s − 6.82·49-s − 10.3·53-s − 16.0·55-s − 6.54·59-s + 3.04·61-s + 1.25·63-s + 8·65-s + 6.54·67-s + 11.3·71-s + 3.27·73-s + ⋯ |
| L(s) = 1 | − 1.36·5-s − 0.158·7-s − 9-s + 1.59·11-s − 0.728·13-s + 0.794·17-s − 0.229·19-s − 0.722·23-s + 0.854·25-s − 0.487·29-s + 1.09·31-s + 0.215·35-s − 1.57·37-s − 0.710·41-s + 0.415·43-s + 1.36·45-s − 0.0610·47-s − 0.974·49-s − 1.42·53-s − 2.16·55-s − 0.852·59-s + 0.389·61-s + 0.158·63-s + 0.992·65-s + 0.800·67-s + 1.34·71-s + 0.383·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9743836572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9743836572\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 3.04T + 5T^{2} \) |
| 7 | \( 1 + 0.418T + 7T^{2} \) |
| 11 | \( 1 - 5.27T + 11T^{2} \) |
| 13 | \( 1 + 2.62T + 13T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 2.62T + 29T^{2} \) |
| 31 | \( 1 - 6.09T + 31T^{2} \) |
| 37 | \( 1 + 9.55T + 37T^{2} \) |
| 41 | \( 1 + 4.54T + 41T^{2} \) |
| 43 | \( 1 - 2.72T + 43T^{2} \) |
| 47 | \( 1 + 0.418T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6.54T + 59T^{2} \) |
| 61 | \( 1 - 3.04T + 61T^{2} \) |
| 67 | \( 1 - 6.54T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 3.27T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−8.153438381344017571522202227302, −7.72342356141037156061633249904, −6.78342774953377429427749108297, −6.27602312300742376224710039965, −5.25807034731074097981823525274, −4.46377791242291841279008734122, −3.61799567668676561165421736135, −3.20552012658871136346728065822, −1.88236790733804721724147846657, −0.53189352930242743753169850583,
0.53189352930242743753169850583, 1.88236790733804721724147846657, 3.20552012658871136346728065822, 3.61799567668676561165421736135, 4.46377791242291841279008734122, 5.25807034731074097981823525274, 6.27602312300742376224710039965, 6.78342774953377429427749108297, 7.72342356141037156061633249904, 8.153438381344017571522202227302
