| L(s) = 1 | + 58.5·3-s − 494.·5-s − 1.33e3·7-s + 1.24e3·9-s − 725.·11-s + 4.82e3·13-s − 2.89e4·15-s − 3.53e3·17-s − 4.49e4·19-s − 7.80e4·21-s + 5.63e4·23-s + 1.66e5·25-s − 5.51e4·27-s − 1.12e4·29-s + 1.44e4·31-s − 4.24e4·33-s + 6.59e5·35-s − 3.80e5·37-s + 2.82e5·39-s + 4.49e5·41-s − 3.31e5·43-s − 6.16e5·45-s − 5.62e5·47-s + 9.51e5·49-s − 2.07e5·51-s + 1.58e6·53-s + 3.58e5·55-s + ⋯ |
| L(s) = 1 | + 1.25·3-s − 1.77·5-s − 1.46·7-s + 0.569·9-s − 0.164·11-s + 0.608·13-s − 2.21·15-s − 0.174·17-s − 1.50·19-s − 1.83·21-s + 0.965·23-s + 2.13·25-s − 0.538·27-s − 0.0852·29-s + 0.0871·31-s − 0.205·33-s + 2.59·35-s − 1.23·37-s + 0.762·39-s + 1.01·41-s − 0.635·43-s − 1.00·45-s − 0.790·47-s + 1.15·49-s − 0.218·51-s + 1.45·53-s + 0.290·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 58.5T + 2.18e3T^{2} \) |
| 5 | \( 1 + 494.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.33e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 725.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.82e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.53e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.49e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.63e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.12e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.44e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.80e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.49e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.31e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.62e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.58e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.72e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.56e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.11e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.99e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.93e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.15e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.33e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.58e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.11e7T + 8.07e13T^{2} \) |
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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
−14.95882033822378570191112440168, −13.37786540821886644298624690090, −12.40018290630151172460195572624, −10.82938234271328333454821691836, −9.084503016451992573952100876503, −8.161955288949685970906861410543, −6.82148741233353065028551239372, −3.96705412880156859380489618705, −3.02212505384605285651582975768, 0,
3.02212505384605285651582975768, 3.96705412880156859380489618705, 6.82148741233353065028551239372, 8.161955288949685970906861410543, 9.084503016451992573952100876503, 10.82938234271328333454821691836, 12.40018290630151172460195572624, 13.37786540821886644298624690090, 14.95882033822378570191112440168
