| L(s) = 1 | − 4.98·2-s + 6.47·3-s + 16.8·4-s − 5·5-s − 32.2·6-s + 19.1·7-s − 43.8·8-s + 14.8·9-s + 24.9·10-s + 11·11-s + 108.·12-s + 55.8·13-s − 95.4·14-s − 32.3·15-s + 83.9·16-s − 116.·17-s − 74.1·18-s − 19·19-s − 84.0·20-s + 124.·21-s − 54.7·22-s − 108.·23-s − 283.·24-s + 25·25-s − 278.·26-s − 78.3·27-s + 322.·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 1.24·3-s + 2.10·4-s − 0.447·5-s − 2.19·6-s + 1.03·7-s − 1.93·8-s + 0.551·9-s + 0.787·10-s + 0.301·11-s + 2.61·12-s + 1.19·13-s − 1.82·14-s − 0.557·15-s + 1.31·16-s − 1.65·17-s − 0.971·18-s − 0.229·19-s − 0.939·20-s + 1.28·21-s − 0.530·22-s − 0.980·23-s − 2.41·24-s + 0.200·25-s − 2.09·26-s − 0.558·27-s + 2.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 4.98T + 8T^{2} \) |
| 3 | \( 1 - 6.47T + 27T^{2} \) |
| 7 | \( 1 - 19.1T + 343T^{2} \) |
| 13 | \( 1 - 55.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 97.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 343.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 164.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 69.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 602.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 254.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 500.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 351.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 435.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 600.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 630.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 761.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 549.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.04e3T + 9.12e5T^{2} \) |
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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
−8.805750976135569924600621498890, −8.445362242652457132404695595114, −7.977980613341703310219017824044, −7.08180753126218623417987055796, −6.18340612255159780409454070311, −4.47830046855886923495578710817, −3.40284470463235332154050249856, −2.13374873713143071871719700583, −1.53670767867273499302280286764, 0,
1.53670767867273499302280286764, 2.13374873713143071871719700583, 3.40284470463235332154050249856, 4.47830046855886923495578710817, 6.18340612255159780409454070311, 7.08180753126218623417987055796, 7.977980613341703310219017824044, 8.445362242652457132404695595114, 8.805750976135569924600621498890
