| L(s) = 1 | + 5.42·2-s + 0.444·3-s + 21.4·4-s + 10.3·5-s + 2.41·6-s + 7·7-s + 72.8·8-s − 26.8·9-s + 56.2·10-s + 9.53·12-s − 8.18·13-s + 37.9·14-s + 4.61·15-s + 223.·16-s − 61.3·17-s − 145.·18-s + 98.0·19-s + 222.·20-s + 3.11·21-s + 181.·23-s + 32.4·24-s − 17.4·25-s − 44.3·26-s − 23.9·27-s + 150.·28-s + 131.·29-s + 25.0·30-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 0.0856·3-s + 2.67·4-s + 0.927·5-s + 0.164·6-s + 0.377·7-s + 3.22·8-s − 0.992·9-s + 1.77·10-s + 0.229·12-s − 0.174·13-s + 0.724·14-s + 0.0794·15-s + 3.49·16-s − 0.875·17-s − 1.90·18-s + 1.18·19-s + 2.48·20-s + 0.0323·21-s + 1.64·23-s + 0.275·24-s − 0.139·25-s − 0.334·26-s − 0.170·27-s + 1.01·28-s + 0.845·29-s + 0.152·30-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
| \(L(2)\) |
\(\approx\) |
\(9.113306856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.113306856\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.42T + 8T^{2} \) |
| 3 | \( 1 - 0.444T + 27T^{2} \) |
| 5 | \( 1 - 10.3T + 125T^{2} \) |
| 13 | \( 1 + 8.18T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 98.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 181.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 131.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 317.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 298.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 46.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 52.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 548.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 52.2T + 1.48e5T^{2} \) |
| 59 | \( 1 + 636.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 766.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 470.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 406.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 867.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 441.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.14e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 353.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.53e3T + 9.12e5T^{2} \) |
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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
−10.11799623249350030886782948507, −8.941412991944402782448180029110, −7.80334187292767620567748342118, −6.71166795969366435336027536072, −6.13314758330019986209888345376, −5.13766083397801409779545946196, −4.75635355079015629104835055030, −3.24666285940320894932831012740, −2.64896602319485925343473610246, −1.49717932884271529545063478804,
1.49717932884271529545063478804, 2.64896602319485925343473610246, 3.24666285940320894932831012740, 4.75635355079015629104835055030, 5.13766083397801409779545946196, 6.13314758330019986209888345376, 6.71166795969366435336027536072, 7.80334187292767620567748342118, 8.941412991944402782448180029110, 10.11799623249350030886782948507
