| L(s) = 1 | − 9·3-s + 12.4·5-s − 98.5·7-s + 81·9-s + 616.·11-s − 169·13-s − 112.·15-s − 1.78e3·17-s − 1.12e3·19-s + 887.·21-s + 4.72e3·23-s − 2.96e3·25-s − 729·27-s + 3.49e3·29-s − 3.90e3·31-s − 5.54e3·33-s − 1.23e3·35-s + 4.64e3·37-s + 1.52e3·39-s + 5.38e3·41-s + 1.19e4·43-s + 1.01e3·45-s + 1.50e4·47-s − 7.08e3·49-s + 1.60e4·51-s − 8.69e3·53-s + 7.70e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.223·5-s − 0.760·7-s + 0.333·9-s + 1.53·11-s − 0.277·13-s − 0.128·15-s − 1.49·17-s − 0.715·19-s + 0.439·21-s + 1.86·23-s − 0.950·25-s − 0.192·27-s + 0.770·29-s − 0.730·31-s − 0.887·33-s − 0.169·35-s + 0.558·37-s + 0.160·39-s + 0.500·41-s + 0.985·43-s + 0.0744·45-s + 0.995·47-s − 0.421·49-s + 0.864·51-s − 0.425·53-s + 0.343·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 13 | \( 1 + 169T \) |
| good | 5 | \( 1 - 12.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 98.5T + 1.68e4T^{2} \) |
| 11 | \( 1 - 616.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.78e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.12e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.72e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.64e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.38e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.50e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.69e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.86e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.24e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.21e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.56e4T + 8.58e9T^{2} \) |
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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
−9.211431575429029623669179303578, −8.948991191808927418551675755753, −7.33080030700433570920110666198, −6.57339924587691663423424612856, −6.01495936880971091665315909631, −4.68802058595333758505855313838, −3.87730203565922456947232277867, −2.51878139657054586214415591907, −1.21346068121881222445521987019, 0,
1.21346068121881222445521987019, 2.51878139657054586214415591907, 3.87730203565922456947232277867, 4.68802058595333758505855313838, 6.01495936880971091665315909631, 6.57339924587691663423424612856, 7.33080030700433570920110666198, 8.948991191808927418551675755753, 9.211431575429029623669179303578
