| L(s) = 1 | − 2-s + 0.303·3-s + 4-s + 0.384·5-s − 0.303·6-s + 3.38·7-s − 8-s − 2.90·9-s − 0.384·10-s − 4.48·11-s + 0.303·12-s + 4.53·13-s − 3.38·14-s + 0.116·15-s + 16-s − 3.64·17-s + 2.90·18-s − 19-s + 0.384·20-s + 1.02·21-s + 4.48·22-s − 6.29·23-s − 0.303·24-s − 4.85·25-s − 4.53·26-s − 1.79·27-s + 3.38·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.175·3-s + 0.5·4-s + 0.171·5-s − 0.124·6-s + 1.28·7-s − 0.353·8-s − 0.969·9-s − 0.121·10-s − 1.35·11-s + 0.0877·12-s + 1.25·13-s − 0.905·14-s + 0.0301·15-s + 0.250·16-s − 0.883·17-s + 0.685·18-s − 0.229·19-s + 0.0858·20-s + 0.224·21-s + 0.955·22-s − 1.31·23-s − 0.0620·24-s − 0.970·25-s − 0.889·26-s − 0.345·27-s + 0.640·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 - 0.303T + 3T^{2} \) |
| 5 | \( 1 - 0.384T + 5T^{2} \) |
| 7 | \( 1 - 3.38T + 7T^{2} \) |
| 11 | \( 1 + 4.48T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 + 3.64T + 17T^{2} \) |
| 23 | \( 1 + 6.29T + 23T^{2} \) |
| 29 | \( 1 + 4.93T + 29T^{2} \) |
| 31 | \( 1 - 3.40T + 31T^{2} \) |
| 37 | \( 1 + 1.43T + 37T^{2} \) |
| 41 | \( 1 + 2.89T + 41T^{2} \) |
| 43 | \( 1 - 1.09T + 43T^{2} \) |
| 47 | \( 1 + 3.58T + 47T^{2} \) |
| 59 | \( 1 + 8.65T + 59T^{2} \) |
| 61 | \( 1 - 0.670T + 61T^{2} \) |
| 67 | \( 1 + 9.02T + 67T^{2} \) |
| 71 | \( 1 - 2.48T + 71T^{2} \) |
| 73 | \( 1 + 0.918T + 73T^{2} \) |
| 79 | \( 1 - 7.94T + 79T^{2} \) |
| 83 | \( 1 - 4.26T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 3.38T + 97T^{2} \) |
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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
−8.560560409587597097088321762925, −8.103935389451522918534364919530, −7.63853369219592350796859910311, −6.30254616766083010543699536905, −5.71255590105778829590305337943, −4.80038946802293195780429468956, −3.64659554048916784300342198307, −2.42883609165526475185658011732, −1.71186186409129015134099055312, 0,
1.71186186409129015134099055312, 2.42883609165526475185658011732, 3.64659554048916784300342198307, 4.80038946802293195780429468956, 5.71255590105778829590305337943, 6.30254616766083010543699536905, 7.63853369219592350796859910311, 8.103935389451522918534364919530, 8.560560409587597097088321762925
