| L(s) = 1 | − 25.1·3-s − 25·5-s − 49·7-s + 390.·9-s − 365.·11-s + 247.·13-s + 629.·15-s + 1.63e3·17-s − 361.·19-s + 1.23e3·21-s + 1.41e3·23-s + 625·25-s − 3.70e3·27-s + 5.20e3·29-s + 5.88e3·31-s + 9.18e3·33-s + 1.22e3·35-s − 1.12e4·37-s − 6.23e3·39-s − 7.24e3·41-s + 742.·43-s − 9.75e3·45-s − 5.16e3·47-s + 2.40e3·49-s − 4.12e4·51-s + 1.60e4·53-s + 9.12e3·55-s + ⋯ |
| L(s) = 1 | − 1.61·3-s − 0.447·5-s − 0.377·7-s + 1.60·9-s − 0.909·11-s + 0.406·13-s + 0.721·15-s + 1.37·17-s − 0.229·19-s + 0.610·21-s + 0.557·23-s + 0.200·25-s − 0.977·27-s + 1.14·29-s + 1.09·31-s + 1.46·33-s + 0.169·35-s − 1.34·37-s − 0.656·39-s − 0.673·41-s + 0.0612·43-s − 0.718·45-s − 0.341·47-s + 0.142·49-s − 2.22·51-s + 0.783·53-s + 0.406·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 25T \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 25.1T + 243T^{2} \) |
| 11 | \( 1 + 365.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 247.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.63e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 361.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.20e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.12e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.24e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 742.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.16e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.60e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.07e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.03e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.64e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.73e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.47e5T + 8.58e9T^{2} \) |
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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
−10.51002532111557516219082569342, −10.10903844233335076000604744729, −8.516876770871211887779564138936, −7.38103072762469866591470527212, −6.40519323468951673165037833518, −5.48977852867457603354689274327, −4.62799616177016725493611562158, −3.15561069952064319220838287622, −1.09797306873370059569989114053, 0,
1.09797306873370059569989114053, 3.15561069952064319220838287622, 4.62799616177016725493611562158, 5.48977852867457603354689274327, 6.40519323468951673165037833518, 7.38103072762469866591470527212, 8.516876770871211887779564138936, 10.10903844233335076000604744729, 10.51002532111557516219082569342
