| L(s) = 1 | + 5.51·2-s + 22.4·4-s + 5·5-s + 79.7·8-s + 27.5·10-s − 34.5·11-s + 68.8·13-s + 260.·16-s + 91.4·17-s + 11.8·19-s + 112.·20-s − 190.·22-s + 0.104·23-s + 25·25-s + 380.·26-s − 190.·29-s + 159.·31-s + 798.·32-s + 504.·34-s − 177.·37-s + 65.2·38-s + 398.·40-s − 145.·41-s + 8.25·43-s − 774.·44-s + 0.574·46-s − 260.·47-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 2.80·4-s + 0.447·5-s + 3.52·8-s + 0.872·10-s − 0.946·11-s + 1.46·13-s + 4.06·16-s + 1.30·17-s + 0.142·19-s + 1.25·20-s − 1.84·22-s + 0.000944·23-s + 0.200·25-s + 2.86·26-s − 1.22·29-s + 0.925·31-s + 4.41·32-s + 2.54·34-s − 0.790·37-s + 0.278·38-s + 1.57·40-s − 0.553·41-s + 0.0292·43-s − 2.65·44-s + 0.00184·46-s − 0.808·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.96933332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.96933332\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 5.51T + 8T^{2} \) |
| 11 | \( 1 + 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 91.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 0.104T + 1.21e4T^{2} \) |
| 29 | \( 1 + 190.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 177.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 8.25T + 7.95e4T^{2} \) |
| 47 | \( 1 + 260.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 353.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 240.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 778.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 151.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 311.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 639.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 391.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 493.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 473.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 839.T + 9.12e5T^{2} \) |
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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
−8.363835170448754283572389860865, −7.68786844051397681431036531713, −6.77893092626830344778664598623, −6.03296467898378840454383364030, −5.46413692874092354570280279476, −4.84351856995033682732403851388, −3.68061146576806708701774357429, −3.23306330096569356934323660889, −2.16717496199725693700868241537, −1.22952304195467700663502412052,
1.22952304195467700663502412052, 2.16717496199725693700868241537, 3.23306330096569356934323660889, 3.68061146576806708701774357429, 4.84351856995033682732403851388, 5.46413692874092354570280279476, 6.03296467898378840454383364030, 6.77893092626830344778664598623, 7.68786844051397681431036531713, 8.363835170448754283572389860865
