L(s) = 1 | + 2-s + 4-s + 3·5-s − 7-s + 8-s − 3·9-s + 3·10-s − 14-s + 16-s + 3·17-s − 3·18-s + 8·19-s + 3·20-s − 23-s + 4·25-s − 28-s − 5·29-s + 8·31-s + 32-s + 3·34-s − 3·35-s − 3·36-s − 5·37-s + 8·38-s + 3·40-s − 5·41-s − 4·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s + 0.353·8-s − 9-s + 0.948·10-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.707·18-s + 1.83·19-s + 0.670·20-s − 0.208·23-s + 4/5·25-s − 0.188·28-s − 0.928·29-s + 1.43·31-s + 0.176·32-s + 0.514·34-s − 0.507·35-s − 1/2·36-s − 0.821·37-s + 1.29·38-s + 0.474·40-s − 0.780·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.181996405\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.181996405\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.20441060810315, −13.90672377071760, −13.40063752005193, −13.27185177618893, −12.27776366321084, −11.94231710649981, −11.59941163729232, −10.80750179639052, −10.26574654898732, −9.819271559972655, −9.386364765212082, −8.809344119755723, −8.063671461251195, −7.584618060408327, −6.735305648559674, −6.440814122286023, −5.634583585184042, −5.437470505782328, −5.053730705596414, −4.017067009266276, −3.277796540902445, −2.954742976222392, −2.218736429343807, −1.538443416852809, −0.6947341640455030,
0.6947341640455030, 1.538443416852809, 2.218736429343807, 2.954742976222392, 3.277796540902445, 4.017067009266276, 5.053730705596414, 5.437470505782328, 5.634583585184042, 6.440814122286023, 6.735305648559674, 7.584618060408327, 8.063671461251195, 8.809344119755723, 9.386364765212082, 9.819271559972655, 10.26574654898732, 10.80750179639052, 11.59941163729232, 11.94231710649981, 12.27776366321084, 13.27185177618893, 13.40063752005193, 13.90672377071760, 14.20441060810315