| L(s) = 1 | − 4.47·2-s − 9·3-s − 11.9·4-s + 56.7·5-s + 40.2·6-s − 133.·7-s + 196.·8-s + 81·9-s − 254.·10-s + 134.·11-s + 107.·12-s + 925.·13-s + 598.·14-s − 511.·15-s − 496.·16-s − 1.78e3·17-s − 362.·18-s + 917.·19-s − 680.·20-s + 1.20e3·21-s − 599.·22-s − 1.35e3·23-s − 1.77e3·24-s + 99.4·25-s − 4.13e3·26-s − 729·27-s + 1.60e3·28-s + ⋯ |
| L(s) = 1 | − 0.790·2-s − 0.577·3-s − 0.374·4-s + 1.01·5-s + 0.456·6-s − 1.03·7-s + 1.08·8-s + 0.333·9-s − 0.803·10-s + 0.333·11-s + 0.216·12-s + 1.51·13-s + 0.816·14-s − 0.586·15-s − 0.485·16-s − 1.49·17-s − 0.263·18-s + 0.583·19-s − 0.380·20-s + 0.595·21-s − 0.264·22-s − 0.532·23-s − 0.627·24-s + 0.0318·25-s − 1.20·26-s − 0.192·27-s + 0.386·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9235410926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9235410926\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 103 | \( 1 - 1.06e4T \) |
| good | 2 | \( 1 + 4.47T + 32T^{2} \) |
| 5 | \( 1 - 56.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 133.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 134.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 925.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.78e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 917.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.35e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.36e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.48e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.74e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.28e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.29e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 9.25e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.74e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.17e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.05e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.43e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.36e4T + 8.58e9T^{2} \) |
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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
−10.52101167314244766231889078752, −9.892872349006429476712305488323, −9.113654609203903819448358602222, −8.320562023266131327202958765131, −6.70504322773585233663234518589, −6.21329013975602786454501719838, −4.93140364799070457573938392965, −3.62962072684351979089642795711, −1.82601921638072566250003461154, −0.64164665595384212634332158541,
0.64164665595384212634332158541, 1.82601921638072566250003461154, 3.62962072684351979089642795711, 4.93140364799070457573938392965, 6.21329013975602786454501719838, 6.70504322773585233663234518589, 8.320562023266131327202958765131, 9.113654609203903819448358602222, 9.892872349006429476712305488323, 10.52101167314244766231889078752
