| L(s) = 1 | + 3·5-s + 7-s − 3·13-s − 4·17-s − 19-s + 4·23-s + 4·25-s + 3·29-s + 2·31-s + 3·35-s − 3·37-s − 4·43-s + 3·47-s + 49-s + 10·53-s + 3·59-s − 2·61-s − 9·65-s − 9·67-s − 16·71-s − 5·73-s + 6·79-s + 16·83-s − 12·85-s + 6·89-s − 3·91-s − 3·95-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s − 0.832·13-s − 0.970·17-s − 0.229·19-s + 0.834·23-s + 4/5·25-s + 0.557·29-s + 0.359·31-s + 0.507·35-s − 0.493·37-s − 0.609·43-s + 0.437·47-s + 1/7·49-s + 1.37·53-s + 0.390·59-s − 0.256·61-s − 1.11·65-s − 1.09·67-s − 1.89·71-s − 0.585·73-s + 0.675·79-s + 1.75·83-s − 1.30·85-s + 0.635·89-s − 0.314·91-s − 0.307·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.52941385876351, −13.47435310424286, −13.03310215218514, −12.28670409025937, −11.92734025189365, −11.40899614274541, −10.68799558370199, −10.36883130197066, −10.03107171299053, −9.287770479541978, −8.995055784356183, −8.580501041308138, −7.847469856490353, −7.259528536183438, −6.793722345881620, −6.283921522108173, −5.781567021635231, −5.147655596122907, −4.798779108810469, −4.245184603016774, −3.410964752237225, −2.617793646767393, −2.325293046304991, −1.666144849588760, −1.001993748125924, 0,
1.001993748125924, 1.666144849588760, 2.325293046304991, 2.617793646767393, 3.410964752237225, 4.245184603016774, 4.798779108810469, 5.147655596122907, 5.781567021635231, 6.283921522108173, 6.793722345881620, 7.259528536183438, 7.847469856490353, 8.580501041308138, 8.995055784356183, 9.287770479541978, 10.03107171299053, 10.36883130197066, 10.68799558370199, 11.40899614274541, 11.92734025189365, 12.28670409025937, 13.03310215218514, 13.47435310424286, 13.52941385876351
