| L(s) = 1 | − 5.92·3-s + 12.8·5-s + 16.9·7-s + 8.14·9-s + 11·11-s − 74.6·13-s − 76.2·15-s − 82.7·17-s + 67.9·19-s − 100.·21-s + 13.3·23-s + 40.2·25-s + 111.·27-s − 168.·29-s − 65.4·31-s − 65.2·33-s + 217.·35-s − 40.8·37-s + 442.·39-s + 274.·41-s + 2.28·43-s + 104.·45-s + 71.8·47-s − 56.4·49-s + 490.·51-s + 149.·53-s + 141.·55-s + ⋯ |
| L(s) = 1 | − 1.14·3-s + 1.14·5-s + 0.914·7-s + 0.301·9-s + 0.301·11-s − 1.59·13-s − 1.31·15-s − 1.18·17-s + 0.820·19-s − 1.04·21-s + 0.121·23-s + 0.322·25-s + 0.796·27-s − 1.08·29-s − 0.379·31-s − 0.343·33-s + 1.05·35-s − 0.181·37-s + 1.81·39-s + 1.04·41-s + 0.00811·43-s + 0.346·45-s + 0.222·47-s − 0.164·49-s + 1.34·51-s + 0.386·53-s + 0.346·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 5.92T + 27T^{2} \) |
| 5 | \( 1 - 12.8T + 125T^{2} \) |
| 7 | \( 1 - 16.9T + 343T^{2} \) |
| 13 | \( 1 + 74.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 82.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 67.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 13.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 168.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 65.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 40.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 274.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 2.28T + 7.95e4T^{2} \) |
| 47 | \( 1 - 71.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 545.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 101.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 411.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 470.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 610.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 978.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 26.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 352.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 847.T + 9.12e5T^{2} \) |
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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
−9.659890211653806100668861196911, −9.021088455503012753060679925998, −7.67320478042031699270768780324, −6.82673914574715326162010760868, −5.82693718417527506000725229542, −5.22613680538853857620486748250, −4.45122371502964288719910280408, −2.55201582993080766996807852719, −1.52242411151265508876516760398, 0,
1.52242411151265508876516760398, 2.55201582993080766996807852719, 4.45122371502964288719910280408, 5.22613680538853857620486748250, 5.82693718417527506000725229542, 6.82673914574715326162010760868, 7.67320478042031699270768780324, 9.021088455503012753060679925998, 9.659890211653806100668861196911
