| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 9.50·5-s − 36·6-s − 69.2·7-s + 64·8-s + 81·9-s − 38.0·10-s + 374.·11-s − 144·12-s − 450.·13-s − 277.·14-s + 85.5·15-s + 256·16-s + 1.13e3·17-s + 324·18-s − 1.93e3·19-s − 152.·20-s + 623.·21-s + 1.49e3·22-s + 2.13e3·23-s − 576·24-s − 3.03e3·25-s − 1.80e3·26-s − 729·27-s − 1.10e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.169·5-s − 0.408·6-s − 0.534·7-s + 0.353·8-s + 0.333·9-s − 0.120·10-s + 0.932·11-s − 0.288·12-s − 0.739·13-s − 0.377·14-s + 0.0981·15-s + 0.250·16-s + 0.950·17-s + 0.235·18-s − 1.23·19-s − 0.0849·20-s + 0.308·21-s + 0.659·22-s + 0.841·23-s − 0.204·24-s − 0.971·25-s − 0.522·26-s − 0.192·27-s − 0.267·28-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 | \( 1 + 9.50T + 3.12e3T^{2} \) |
| 7 | \( 1 + 69.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 374.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 450.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.13e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.93e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.13e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.40e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 640.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 206.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.06e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.67e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.02e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 2.60e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.10e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.50e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.49e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.15e4T + 8.58e9T^{2} \) |
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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
−10.28625351281267292353525775911, −9.486534790252677668124437091488, −8.144444347058944164023822086404, −6.94225537545125052183869980807, −6.31006949437139334543208993925, −5.20443081880767247582420192745, −4.19411013892117047538255936959, −3.11885275600258130289856156106, −1.57519226941104191984886677935, 0,
1.57519226941104191984886677935, 3.11885275600258130289856156106, 4.19411013892117047538255936959, 5.20443081880767247582420192745, 6.31006949437139334543208993925, 6.94225537545125052183869980807, 8.144444347058944164023822086404, 9.486534790252677668124437091488, 10.28625351281267292353525775911
