| L(s) = 1 | + 3-s + 2·5-s − 4·7-s − 2·9-s − 3·11-s + 4·13-s + 2·15-s + 7·17-s + 19-s − 4·21-s − 8·23-s − 25-s − 5·27-s − 6·29-s + 9·31-s − 3·33-s − 8·35-s − 4·37-s + 4·39-s + 6·41-s − 11·43-s − 4·45-s + 4·47-s + 9·49-s + 7·51-s − 53-s − 6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 1.51·7-s − 2/3·9-s − 0.904·11-s + 1.10·13-s + 0.516·15-s + 1.69·17-s + 0.229·19-s − 0.872·21-s − 1.66·23-s − 1/5·25-s − 0.962·27-s − 1.11·29-s + 1.61·31-s − 0.522·33-s − 1.35·35-s − 0.657·37-s + 0.640·39-s + 0.937·41-s − 1.67·43-s − 0.596·45-s + 0.583·47-s + 9/7·49-s + 0.980·51-s − 0.137·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.073816552893148095385155950772, −7.53348796035411653018805013218, −6.31079814697090662624007906533, −5.92326974206542694350000426707, −5.43854837715442909336402900255, −3.96155239487756496248072678828, −3.21719831863771850678009370189, −2.70314782691968293692611427899, −1.56903682747979920584027851365, 0,
1.56903682747979920584027851365, 2.70314782691968293692611427899, 3.21719831863771850678009370189, 3.96155239487756496248072678828, 5.43854837715442909336402900255, 5.92326974206542694350000426707, 6.31079814697090662624007906533, 7.53348796035411653018805013218, 8.073816552893148095385155950772
