| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 28.1·5-s + 36·6-s − 61.0·7-s + 64·8-s + 81·9-s − 112.·10-s − 67.1·11-s + 144·12-s − 428.·13-s − 244.·14-s − 253.·15-s + 256·16-s + 159.·17-s + 324·18-s + 374.·19-s − 451.·20-s − 549.·21-s − 268.·22-s − 1.39e3·23-s + 576·24-s − 2.32e3·25-s − 1.71e3·26-s + 729·27-s − 976.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.504·5-s + 0.408·6-s − 0.470·7-s + 0.353·8-s + 0.333·9-s − 0.356·10-s − 0.167·11-s + 0.288·12-s − 0.702·13-s − 0.332·14-s − 0.291·15-s + 0.250·16-s + 0.133·17-s + 0.235·18-s + 0.237·19-s − 0.252·20-s − 0.271·21-s − 0.118·22-s − 0.548·23-s + 0.204·24-s − 0.745·25-s − 0.497·26-s + 0.192·27-s − 0.235·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 - 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 5 | \( 1 + 28.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 61.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 67.1T + 1.61e5T^{2} \) |
| 13 | \( 1 + 428.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 159.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 374.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.39e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.25e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.16e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.20e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.35e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.94e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.84e3T + 4.18e8T^{2} \) |
| 61 | \( 1 - 6.66T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.12e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.71e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.27e4T + 8.58e9T^{2} \) |
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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
−10.12089120333615676970870117474, −9.370477574138385731903296005760, −8.046825772020272061586401305564, −7.39309165973029272066502996349, −6.28287788015314149659621105110, −5.09505553484354848377631603697, −3.95964335513855308948421984889, −3.10247799184704530040312835372, −1.87550746782260636163026772584, 0,
1.87550746782260636163026772584, 3.10247799184704530040312835372, 3.95964335513855308948421984889, 5.09505553484354848377631603697, 6.28287788015314149659621105110, 7.39309165973029272066502996349, 8.046825772020272061586401305564, 9.370477574138385731903296005760, 10.12089120333615676970870117474
