| L(s) = 1 | − 3·3-s − 2·7-s + 6·9-s + 11-s − 4·13-s + 5·17-s + 19-s + 6·21-s − 9·27-s − 8·29-s − 10·31-s − 3·33-s − 6·37-s + 12·39-s − 3·41-s − 4·43-s + 4·47-s − 3·49-s − 15·51-s + 6·53-s − 3·57-s − 8·59-s − 10·61-s − 12·63-s + 67-s + 12·71-s − 3·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.755·7-s + 2·9-s + 0.301·11-s − 1.10·13-s + 1.21·17-s + 0.229·19-s + 1.30·21-s − 1.73·27-s − 1.48·29-s − 1.79·31-s − 0.522·33-s − 0.986·37-s + 1.92·39-s − 0.468·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 2.10·51-s + 0.824·53-s − 0.397·57-s − 1.04·59-s − 1.28·61-s − 1.51·63-s + 0.122·67-s + 1.42·71-s − 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.34657412976956, −12.74801283978489, −12.33080860911268, −12.21067280679457, −11.68648489725893, −11.14399703123589, −10.70958942370824, −10.30034925392797, −9.826056944914093, −9.332690362646847, −9.110605672499082, −8.072564320183188, −7.510722580201992, −7.217635313766747, −6.721250598350825, −6.208630219189784, −5.705594382376640, −5.205387583027578, −5.071083779621779, −4.241288694540976, −3.570448344492521, −3.321648176935611, −2.264091181558236, −1.656180069187695, −1.004804763332171, 0, 0,
1.004804763332171, 1.656180069187695, 2.264091181558236, 3.321648176935611, 3.570448344492521, 4.241288694540976, 5.071083779621779, 5.205387583027578, 5.705594382376640, 6.208630219189784, 6.721250598350825, 7.217635313766747, 7.510722580201992, 8.072564320183188, 9.110605672499082, 9.332690362646847, 9.826056944914093, 10.30034925392797, 10.70958942370824, 11.14399703123589, 11.68648489725893, 12.21067280679457, 12.33080860911268, 12.74801283978489, 13.34657412976956
