| L(s) = 1 | − 4.15·2-s − 9·3-s − 14.7·4-s − 37.5·5-s + 37.3·6-s + 76.4·7-s + 194.·8-s + 81·9-s + 155.·10-s + 132.·12-s − 169.·13-s − 317.·14-s + 337.·15-s − 333.·16-s + 0.875·17-s − 336.·18-s + 817.·19-s + 553.·20-s − 688.·21-s + 749.·23-s − 1.74e3·24-s − 1.71e3·25-s + 704.·26-s − 729·27-s − 1.12e3·28-s − 6.04e3·29-s − 1.40e3·30-s + ⋯ |
| L(s) = 1 | − 0.733·2-s − 0.577·3-s − 0.461·4-s − 0.671·5-s + 0.423·6-s + 0.589·7-s + 1.07·8-s + 0.333·9-s + 0.492·10-s + 0.266·12-s − 0.278·13-s − 0.433·14-s + 0.387·15-s − 0.325·16-s + 0.000735·17-s − 0.244·18-s + 0.519·19-s + 0.309·20-s − 0.340·21-s + 0.295·23-s − 0.619·24-s − 0.549·25-s + 0.204·26-s − 0.192·27-s − 0.272·28-s − 1.33·29-s − 0.284·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5840214328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5840214328\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 4.15T + 32T^{2} \) |
| 5 | \( 1 + 37.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 76.4T + 1.68e4T^{2} \) |
| 13 | \( 1 + 169.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 0.875T + 1.41e6T^{2} \) |
| 19 | \( 1 - 817.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 749.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.58e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.62e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.82e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.28e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.16e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.75e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.46e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−10.62747249996840682021436115910, −9.637403593532933429531782819135, −8.784367233335181981984995726319, −7.76645257172123051135918296421, −7.21177792217797386139974358204, −5.61746914027455304337579506837, −4.70672983249106871783321159994, −3.70420307804801337837294864871, −1.72965327582353802102100806118, −0.48968916075945412115532132588,
0.48968916075945412115532132588, 1.72965327582353802102100806118, 3.70420307804801337837294864871, 4.70672983249106871783321159994, 5.61746914027455304337579506837, 7.21177792217797386139974358204, 7.76645257172123051135918296421, 8.784367233335181981984995726319, 9.637403593532933429531782819135, 10.62747249996840682021436115910
