| L(s) = 1 | − 4.60·2-s + 2.77·3-s + 13.1·4-s + 1.84·5-s − 12.7·6-s + 7·7-s − 23.9·8-s − 19.3·9-s − 8.49·10-s + 36.6·12-s − 24.6·13-s − 32.2·14-s + 5.11·15-s + 4.57·16-s − 17.8·17-s + 88.8·18-s − 32.1·19-s + 24.3·20-s + 19.4·21-s + 14.1·23-s − 66.3·24-s − 121.·25-s + 113.·26-s − 128.·27-s + 92.3·28-s + 41.5·29-s − 23.5·30-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 0.533·3-s + 1.64·4-s + 0.164·5-s − 0.868·6-s + 0.377·7-s − 1.05·8-s − 0.714·9-s − 0.268·10-s + 0.880·12-s − 0.525·13-s − 0.615·14-s + 0.0880·15-s + 0.0714·16-s − 0.255·17-s + 1.16·18-s − 0.388·19-s + 0.272·20-s + 0.201·21-s + 0.128·23-s − 0.564·24-s − 0.972·25-s + 0.855·26-s − 0.915·27-s + 0.623·28-s + 0.266·29-s − 0.143·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8910917653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8910917653\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 4.60T + 8T^{2} \) |
| 3 | \( 1 - 2.77T + 27T^{2} \) |
| 5 | \( 1 - 1.84T + 125T^{2} \) |
| 13 | \( 1 + 24.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 17.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 32.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 14.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 292.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 154.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 277.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 52.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 82.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 712.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 647.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 260.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 369.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 488.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 548.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 105.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.36e3T + 9.12e5T^{2} \) |
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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
−9.714564705290983028211318002806, −8.863172830853683714427373089204, −8.290051248296488210474593081697, −7.64762579511259772465747958526, −6.70667597428426725583791059024, −5.66151317764648870832740442579, −4.29988367753596253747982693244, −2.75232050615603012180054495817, −2.00032933383020723923421643628, −0.63312203665333486677318917132,
0.63312203665333486677318917132, 2.00032933383020723923421643628, 2.75232050615603012180054495817, 4.29988367753596253747982693244, 5.66151317764648870832740442579, 6.70667597428426725583791059024, 7.64762579511259772465747958526, 8.290051248296488210474593081697, 8.863172830853683714427373089204, 9.714564705290983028211318002806
