| L(s) = 1 | + 2·2-s + 2·4-s + 7-s − 4·13-s + 2·14-s − 4·16-s − 5·17-s − 3·19-s − 9·23-s − 8·26-s + 2·28-s − 5·29-s + 2·31-s − 8·32-s − 10·34-s − 4·37-s − 6·38-s + 2·41-s + 43-s − 18·46-s + 49-s − 8·52-s − 3·53-s − 10·58-s + 5·59-s + 5·61-s + 4·62-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.377·7-s − 1.10·13-s + 0.534·14-s − 16-s − 1.21·17-s − 0.688·19-s − 1.87·23-s − 1.56·26-s + 0.377·28-s − 0.928·29-s + 0.359·31-s − 1.41·32-s − 1.71·34-s − 0.657·37-s − 0.973·38-s + 0.312·41-s + 0.152·43-s − 2.65·46-s + 1/7·49-s − 1.10·52-s − 0.412·53-s − 1.31·58-s + 0.650·59-s + 0.640·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.58403053516979, −13.13461068160555, −12.72668219892188, −12.19588850169549, −11.87102647623501, −11.46430954566132, −10.90956559529712, −10.42300928901635, −9.853391528098010, −9.371865198094270, −8.750843813427269, −8.363696780507465, −7.691306179506501, −7.189953070802682, −6.680685771659712, −6.229010822756253, −5.557924805596614, −5.381072323901816, −4.541236810970369, −4.209936194878338, −4.016934185558411, −3.067764145672456, −2.571786782778259, −2.073712652267975, −1.545177606429121, 0, 0,
1.545177606429121, 2.073712652267975, 2.571786782778259, 3.067764145672456, 4.016934185558411, 4.209936194878338, 4.541236810970369, 5.381072323901816, 5.557924805596614, 6.229010822756253, 6.680685771659712, 7.189953070802682, 7.691306179506501, 8.363696780507465, 8.750843813427269, 9.371865198094270, 9.853391528098010, 10.42300928901635, 10.90956559529712, 11.46430954566132, 11.87102647623501, 12.19588850169549, 12.72668219892188, 13.13461068160555, 13.58403053516979
