| L(s) = 1 | − 1.73·2-s + 0.999·4-s − 2·7-s + 1.73·8-s + 11-s − 5.46·13-s + 3.46·14-s − 5·16-s + 5.46·19-s − 1.73·22-s + 6.92·23-s + 9.46·26-s − 1.99·28-s + 3.46·29-s − 10.9·31-s + 5.19·32-s + 4.92·37-s − 9.46·38-s − 3.46·41-s + 4.92·43-s + 0.999·44-s − 11.9·46-s − 6.92·47-s − 3·49-s − 5.46·52-s + 0.928·53-s − 3.46·56-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s − 0.755·7-s + 0.612·8-s + 0.301·11-s − 1.51·13-s + 0.925·14-s − 1.25·16-s + 1.25·19-s − 0.369·22-s + 1.44·23-s + 1.85·26-s − 0.377·28-s + 0.643·29-s − 1.96·31-s + 0.918·32-s + 0.810·37-s − 1.53·38-s − 0.541·41-s + 0.751·43-s + 0.150·44-s − 1.76·46-s − 1.01·47-s − 0.428·49-s − 0.757·52-s + 0.127·53-s − 0.462·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 0.928T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 8.39T + 73T^{2} \) |
| 79 | \( 1 + 6.53T + 79T^{2} \) |
| 83 | \( 1 - 8.53T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.814127321635745912038140176294, −7.69735998848652636947937930134, −7.28546833628206220907980490811, −6.60398367550682932116583418422, −5.37418924240577003230373375310, −4.67505474537592244856044624651, −3.44657967038959692208149858925, −2.48829905494733104436189766369, −1.21743386798162512038033609400, 0,
1.21743386798162512038033609400, 2.48829905494733104436189766369, 3.44657967038959692208149858925, 4.67505474537592244856044624651, 5.37418924240577003230373375310, 6.60398367550682932116583418422, 7.28546833628206220907980490811, 7.69735998848652636947937930134, 8.814127321635745912038140176294
