| L(s) = 1 | − 12.7·3-s − 25·5-s − 68.7·7-s − 80.8·9-s + 327.·11-s + 719.·13-s + 318.·15-s − 379.·17-s + 1.02e3·19-s + 875.·21-s + 779.·23-s + 625·25-s + 4.12e3·27-s − 1.39e3·29-s − 2.74e3·31-s − 4.16e3·33-s + 1.71e3·35-s − 1.26e4·37-s − 9.15e3·39-s + 8.21e3·41-s + 2.25e4·43-s + 2.02e3·45-s − 7.73e3·47-s − 1.20e4·49-s + 4.83e3·51-s + 2.40e3·53-s − 8.18e3·55-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 0.447·5-s − 0.530·7-s − 0.332·9-s + 0.815·11-s + 1.18·13-s + 0.365·15-s − 0.318·17-s + 0.654·19-s + 0.433·21-s + 0.307·23-s + 0.200·25-s + 1.08·27-s − 0.307·29-s − 0.512·31-s − 0.666·33-s + 0.237·35-s − 1.51·37-s − 0.964·39-s + 0.762·41-s + 1.85·43-s + 0.148·45-s − 0.511·47-s − 0.718·49-s + 0.260·51-s + 0.117·53-s − 0.364·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| good | 3 | \( 1 + 12.7T + 243T^{2} \) |
| 7 | \( 1 + 68.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 327.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 719.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 379.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.02e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 779.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.74e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.26e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.21e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.25e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.73e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.40e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.57e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 9.00e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.58e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.31e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.45e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.40e4T + 8.58e9T^{2} \) |
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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
−10.62987837960466234569276205775, −9.313796199708606407184527736847, −8.574194920765514380718969374936, −7.24348414993476455209093194515, −6.31467427659840304162269792085, −5.52541761282739948731512021685, −4.17105650498536724799991153446, −3.12846535711795993808237160403, −1.23616585723922726993668273931, 0,
1.23616585723922726993668273931, 3.12846535711795993808237160403, 4.17105650498536724799991153446, 5.52541761282739948731512021685, 6.31467427659840304162269792085, 7.24348414993476455209093194515, 8.574194920765514380718969374936, 9.313796199708606407184527736847, 10.62987837960466234569276205775
