| L(s) = 1 | + 5.77·3-s − 5·5-s + 7·7-s + 6.31·9-s − 6.22·11-s − 52.5·13-s − 28.8·15-s − 71.2·17-s − 59.5·19-s + 40.4·21-s − 61.3·23-s + 25·25-s − 119.·27-s + 162.·29-s − 91.6·31-s − 35.9·33-s − 35·35-s + 148.·37-s − 303.·39-s − 140.·41-s − 210.·43-s − 31.5·45-s − 98.2·47-s + 49·49-s − 411.·51-s + 87.0·53-s + 31.1·55-s + ⋯ |
| L(s) = 1 | + 1.11·3-s − 0.447·5-s + 0.377·7-s + 0.233·9-s − 0.170·11-s − 1.12·13-s − 0.496·15-s − 1.01·17-s − 0.718·19-s + 0.419·21-s − 0.556·23-s + 0.200·25-s − 0.850·27-s + 1.04·29-s − 0.530·31-s − 0.189·33-s − 0.169·35-s + 0.661·37-s − 1.24·39-s − 0.537·41-s − 0.745·43-s − 0.104·45-s − 0.304·47-s + 0.142·49-s − 1.12·51-s + 0.225·53-s + 0.0763·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 - 5.77T + 27T^{2} \) |
| 11 | \( 1 + 6.22T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 71.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 59.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 61.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 91.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 148.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 140.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 210.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 98.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 87.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 734.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 83.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 79.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 211.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 213.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 517.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.34e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.86e3T + 9.12e5T^{2} \) |
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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
−9.774329938542623185088757098590, −8.875512983146597896112131347947, −8.197503178125327577954562084434, −7.48833513756847185227998788075, −6.43444094008622001560087294065, −4.98357693586686153095302738109, −4.09182737630779287535521394473, −2.87827625179097000567287514037, −1.99366228890928523755141594389, 0,
1.99366228890928523755141594389, 2.87827625179097000567287514037, 4.09182737630779287535521394473, 4.98357693586686153095302738109, 6.43444094008622001560087294065, 7.48833513756847185227998788075, 8.197503178125327577954562084434, 8.875512983146597896112131347947, 9.774329938542623185088757098590
