L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 3·11-s + 16-s + 2·17-s + 2·19-s − 20-s − 3·22-s − 23-s + 25-s − 5·29-s − 5·31-s − 32-s − 2·34-s + 3·37-s − 2·38-s + 40-s + 2·41-s − 9·43-s + 3·44-s + 46-s + 7·47-s − 7·49-s − 50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.904·11-s + 1/4·16-s + 0.485·17-s + 0.458·19-s − 0.223·20-s − 0.639·22-s − 0.208·23-s + 1/5·25-s − 0.928·29-s − 0.898·31-s − 0.176·32-s − 0.342·34-s + 0.493·37-s − 0.324·38-s + 0.158·40-s + 0.312·41-s − 1.37·43-s + 0.452·44-s + 0.147·46-s + 1.02·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.31233506399072, −16.01114088322718, −15.10725258794676, −14.81071094395766, −14.28369706234514, −13.47214486907847, −12.95323075304250, −12.09370449883529, −11.83745781748094, −11.26372505737136, −10.61747291722296, −10.04183255216313, −9.324487658504191, −9.002078096911624, −8.268315813608656, −7.603288138676550, −7.208142474086953, −6.474767092577919, −5.806959108689862, −5.128750177519424, −4.144962096089961, −3.611035293542201, −2.833471294508522, −1.821061540493642, −1.100541245744377, 0,
1.100541245744377, 1.821061540493642, 2.833471294508522, 3.611035293542201, 4.144962096089961, 5.128750177519424, 5.806959108689862, 6.474767092577919, 7.208142474086953, 7.603288138676550, 8.268315813608656, 9.002078096911624, 9.324487658504191, 10.04183255216313, 10.61747291722296, 11.26372505737136, 11.83745781748094, 12.09370449883529, 12.95323075304250, 13.47214486907847, 14.28369706234514, 14.81071094395766, 15.10725258794676, 16.01114088322718, 16.31233506399072