| L(s) = 1 | − 2·5-s + 7-s − 5·13-s − 6·17-s + 8·19-s − 23-s − 25-s + 5·29-s + 5·31-s − 2·35-s − 8·37-s + 3·41-s + 43-s + 2·47-s + 49-s + 10·53-s − 3·59-s + 3·61-s + 10·65-s + 13·67-s − 15·71-s − 12·73-s − 8·79-s − 11·83-s + 12·85-s − 15·89-s − 5·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 1.38·13-s − 1.45·17-s + 1.83·19-s − 0.208·23-s − 1/5·25-s + 0.928·29-s + 0.898·31-s − 0.338·35-s − 1.31·37-s + 0.468·41-s + 0.152·43-s + 0.291·47-s + 1/7·49-s + 1.37·53-s − 0.390·59-s + 0.384·61-s + 1.24·65-s + 1.58·67-s − 1.78·71-s − 1.40·73-s − 0.900·79-s − 1.20·83-s + 1.30·85-s − 1.58·89-s − 0.524·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.38875252971189, −14.14101045258278, −13.64691159227608, −12.99078611854161, −12.40436352970339, −11.88013311372616, −11.59789483000676, −11.24910211454873, −10.26728308799274, −10.13140870673626, −9.435405561477857, −8.709355534675475, −8.463541476135215, −7.646959388802785, −7.273313624141618, −6.984423949731550, −6.111515728938848, −5.434874923207817, −4.853465844626477, −4.409660109685256, −3.859109661263252, −2.990737052638652, −2.563172535082158, −1.731990284992748, −0.8012203782857740, 0,
0.8012203782857740, 1.731990284992748, 2.563172535082158, 2.990737052638652, 3.859109661263252, 4.409660109685256, 4.853465844626477, 5.434874923207817, 6.111515728938848, 6.984423949731550, 7.273313624141618, 7.646959388802785, 8.463541476135215, 8.709355534675475, 9.435405561477857, 10.13140870673626, 10.26728308799274, 11.24910211454873, 11.59789483000676, 11.88013311372616, 12.40436352970339, 12.99078611854161, 13.64691159227608, 14.14101045258278, 14.38875252971189
