| L(s) = 1 | − 0.523·2-s − 3-s − 1.72·4-s + 0.523·6-s + 3.20·7-s + 1.95·8-s + 9-s + 2.20·11-s + 1.72·12-s − 3.04·13-s − 1.67·14-s + 2.42·16-s − 0.523·18-s + 19-s − 3.20·21-s − 1.15·22-s − 3.24·23-s − 1.95·24-s + 1.59·26-s − 27-s − 5.52·28-s + 3·29-s − 1.24·31-s − 5.17·32-s − 2.20·33-s − 1.72·36-s + 3.45·37-s + ⋯ |
| L(s) = 1 | − 0.370·2-s − 0.577·3-s − 0.862·4-s + 0.213·6-s + 1.21·7-s + 0.690·8-s + 0.333·9-s + 0.663·11-s + 0.498·12-s − 0.845·13-s − 0.448·14-s + 0.607·16-s − 0.123·18-s + 0.229·19-s − 0.698·21-s − 0.245·22-s − 0.677·23-s − 0.398·24-s + 0.313·26-s − 0.192·27-s − 1.04·28-s + 0.557·29-s − 0.224·31-s − 0.915·32-s − 0.383·33-s − 0.287·36-s + 0.567·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.022219076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.022219076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.523T + 2T^{2} \) |
| 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 - 2.20T + 11T^{2} \) |
| 13 | \( 1 + 3.04T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 - 3.45T + 37T^{2} \) |
| 41 | \( 1 - 8.55T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 3.35T + 47T^{2} \) |
| 53 | \( 1 + 0.904T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 3.40T + 61T^{2} \) |
| 67 | \( 1 - 2.29T + 67T^{2} \) |
| 71 | \( 1 - 7.29T + 71T^{2} \) |
| 73 | \( 1 + 1.09T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 8.20T + 83T^{2} \) |
| 89 | \( 1 + 7.40T + 89T^{2} \) |
| 97 | \( 1 - 7.14T + 97T^{2} \) |
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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
−9.629860879845362478439221425537, −8.716679215621247763113223089675, −7.993682589071579310096214681055, −7.32195574012172670531920200749, −6.20821438836100704436769410647, −5.14301442617203784710302517878, −4.67628703768186632111555329034, −3.75734166513923852693874690866, −2.01345013425058230689092793958, −0.829674492743556046125623592823,
0.829674492743556046125623592823, 2.01345013425058230689092793958, 3.75734166513923852693874690866, 4.67628703768186632111555329034, 5.14301442617203784710302517878, 6.20821438836100704436769410647, 7.32195574012172670531920200749, 7.993682589071579310096214681055, 8.716679215621247763113223089675, 9.629860879845362478439221425537
