| L(s) = 1 | − 16.8·3-s − 75.2·5-s − 225.·7-s + 40.5·9-s − 121·11-s − 455.·13-s + 1.26e3·15-s + 190.·17-s + 135.·19-s + 3.79e3·21-s + 2.79e3·23-s + 2.53e3·25-s + 3.40e3·27-s + 2.60e3·29-s − 1.05e3·31-s + 2.03e3·33-s + 1.69e4·35-s − 1.25e4·37-s + 7.66e3·39-s + 1.13e3·41-s + 1.46e4·43-s − 3.05e3·45-s − 1.68e4·47-s + 3.40e4·49-s − 3.21e3·51-s − 3.31e3·53-s + 9.10e3·55-s + ⋯ |
| L(s) = 1 | − 1.08·3-s − 1.34·5-s − 1.73·7-s + 0.166·9-s − 0.301·11-s − 0.747·13-s + 1.45·15-s + 0.160·17-s + 0.0860·19-s + 1.87·21-s + 1.10·23-s + 0.810·25-s + 0.899·27-s + 0.575·29-s − 0.197·31-s + 0.325·33-s + 2.34·35-s − 1.50·37-s + 0.807·39-s + 0.104·41-s + 1.21·43-s − 0.224·45-s − 1.11·47-s + 2.02·49-s − 0.172·51-s − 0.162·53-s + 0.405·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 + 16.8T + 243T^{2} \) |
| 5 | \( 1 + 75.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 225.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 455.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 190.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 135.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.79e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.60e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.25e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.13e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.46e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.68e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.31e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.57e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.22e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.30e4T + 8.58e9T^{2} \) |
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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
−9.368161914253458161922791752693, −8.336433965301398899360518761341, −7.18914064583447802869998502677, −6.71869716718249065247647314984, −5.66697877356769513908442403262, −4.75742619527773473140276697917, −3.60592561363387118896918645495, −2.82449104451948099542066764259, −0.66667213645716194713569473760, 0,
0.66667213645716194713569473760, 2.82449104451948099542066764259, 3.60592561363387118896918645495, 4.75742619527773473140276697917, 5.66697877356769513908442403262, 6.71869716718249065247647314984, 7.18914064583447802869998502677, 8.336433965301398899360518761341, 9.368161914253458161922791752693
