| L(s) = 1 | + 2.65·2-s − 3·3-s − 0.924·4-s − 15.5·5-s − 7.97·6-s − 9.37·7-s − 23.7·8-s + 9·9-s − 41.4·10-s + 11·11-s + 2.77·12-s − 51.4·13-s − 24.9·14-s + 46.7·15-s − 55.7·16-s + 41.6·17-s + 23.9·18-s + 130.·19-s + 14.4·20-s + 28.1·21-s + 29.2·22-s + 208.·23-s + 71.2·24-s + 118.·25-s − 136.·26-s − 27·27-s + 8.66·28-s + ⋯ |
| L(s) = 1 | + 0.940·2-s − 0.577·3-s − 0.115·4-s − 1.39·5-s − 0.542·6-s − 0.506·7-s − 1.04·8-s + 0.333·9-s − 1.31·10-s + 0.301·11-s + 0.0667·12-s − 1.09·13-s − 0.476·14-s + 0.805·15-s − 0.871·16-s + 0.593·17-s + 0.313·18-s + 1.57·19-s + 0.161·20-s + 0.292·21-s + 0.283·22-s + 1.88·23-s + 0.605·24-s + 0.945·25-s − 1.03·26-s − 0.192·27-s + 0.0584·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 2.65T + 8T^{2} \) |
| 5 | \( 1 + 15.5T + 125T^{2} \) |
| 7 | \( 1 + 9.37T + 343T^{2} \) |
| 13 | \( 1 + 51.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 41.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 208.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 131.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 167.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 437.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 37.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 38.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 438.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 525.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 174.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 141.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 149.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 60.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 766.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 756.T + 9.12e5T^{2} \) |
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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
−8.235085864998507815460048150859, −7.42537707786568396311013859154, −6.81885906904070765504228906146, −5.81529640583090321620530499793, −4.90725969828516771789231215367, −4.54137058224696898188587840362, −3.36512494420034458437855646818, −3.04185953305809348466987301316, −0.950292725931284140020841165824, 0,
0.950292725931284140020841165824, 3.04185953305809348466987301316, 3.36512494420034458437855646818, 4.54137058224696898188587840362, 4.90725969828516771789231215367, 5.81529640583090321620530499793, 6.81885906904070765504228906146, 7.42537707786568396311013859154, 8.235085864998507815460048150859
