| L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s + 11-s + 14-s − 16-s − 2·17-s − 4·19-s − 22-s − 4·23-s − 5·25-s + 28-s + 4·29-s + 4·31-s − 5·32-s + 2·34-s − 8·37-s + 4·38-s − 10·41-s − 44-s + 4·46-s + 2·47-s + 49-s + 5·50-s − 6·53-s − 3·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s + 0.301·11-s + 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.213·22-s − 0.834·23-s − 25-s + 0.188·28-s + 0.742·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s − 1.31·37-s + 0.648·38-s − 1.56·41-s − 0.150·44-s + 0.589·46-s + 0.291·47-s + 1/7·49-s + 0.707·50-s − 0.824·53-s − 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.05102055126686, −13.59095481337308, −13.22082151976443, −12.58037730065312, −12.13990547310616, −11.69216695972500, −11.01274166083254, −10.47595812155015, −10.14046424186230, −9.694560274393575, −9.181233666756725, −8.573732768999490, −8.383708699402440, −7.793303687992666, −7.175611429111002, −6.642201027945867, −6.150996645978453, −5.580975970939335, −4.732021333784515, −4.507812121046574, −3.785763909775635, −3.323594767036223, −2.406827095278323, −1.783361481144763, −1.205423211255894, 0, 0,
1.205423211255894, 1.783361481144763, 2.406827095278323, 3.323594767036223, 3.785763909775635, 4.507812121046574, 4.732021333784515, 5.580975970939335, 6.150996645978453, 6.642201027945867, 7.175611429111002, 7.793303687992666, 8.383708699402440, 8.573732768999490, 9.181233666756725, 9.694560274393575, 10.14046424186230, 10.47595812155015, 11.01274166083254, 11.69216695972500, 12.13990547310616, 12.58037730065312, 13.22082151976443, 13.59095481337308, 14.05102055126686
