| L(s) = 1 | + 10·3-s − 84·5-s − 143·9-s + 336·11-s − 584·13-s − 840·15-s + 1.45e3·17-s + 470·19-s + 4.20e3·23-s + 3.93e3·25-s − 3.86e3·27-s + 4.86e3·29-s − 7.37e3·31-s + 3.36e3·33-s + 1.43e4·37-s − 5.84e3·39-s − 6.22e3·41-s − 3.70e3·43-s + 1.20e4·45-s − 1.81e3·47-s + 1.45e4·51-s − 3.72e4·53-s − 2.82e4·55-s + 4.70e3·57-s + 3.43e4·59-s − 2.44e4·61-s + 4.90e4·65-s + ⋯ |
| L(s) = 1 | + 0.641·3-s − 1.50·5-s − 0.588·9-s + 0.837·11-s − 0.958·13-s − 0.963·15-s + 1.22·17-s + 0.298·19-s + 1.65·23-s + 1.25·25-s − 1.01·27-s + 1.07·29-s − 1.37·31-s + 0.537·33-s + 1.72·37-s − 0.614·39-s − 0.578·41-s − 0.305·43-s + 0.884·45-s − 0.119·47-s + 0.784·51-s − 1.82·53-s − 1.25·55-s + 0.191·57-s + 1.28·59-s − 0.842·61-s + 1.44·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 10 T + p^{5} T^{2} \) |
| 5 | \( 1 + 84 T + p^{5} T^{2} \) |
| 11 | \( 1 - 336 T + p^{5} T^{2} \) |
| 13 | \( 1 + 584 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1458 T + p^{5} T^{2} \) |
| 19 | \( 1 - 470 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4200 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4866 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7372 T + p^{5} T^{2} \) |
| 37 | \( 1 - 14330 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6222 T + p^{5} T^{2} \) |
| 43 | \( 1 + 3704 T + p^{5} T^{2} \) |
| 47 | \( 1 + 1812 T + p^{5} T^{2} \) |
| 53 | \( 1 + 37242 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34302 T + p^{5} T^{2} \) |
| 61 | \( 1 + 24476 T + p^{5} T^{2} \) |
| 67 | \( 1 - 17452 T + p^{5} T^{2} \) |
| 71 | \( 1 + 28224 T + p^{5} T^{2} \) |
| 73 | \( 1 + 3602 T + p^{5} T^{2} \) |
| 79 | \( 1 + 42872 T + p^{5} T^{2} \) |
| 83 | \( 1 + 35202 T + p^{5} T^{2} \) |
| 89 | \( 1 + 26730 T + p^{5} T^{2} \) |
| 97 | \( 1 - 16978 T + p^{5} T^{2} \) |
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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
−9.032651152314828871476961988173, −8.151907807435224999155651612292, −7.58358075550629032868440981878, −6.78610692388255677658014028333, −5.41160569603738275873205852152, −4.41408406214326022029159289132, −3.42335796159392902091704565607, −2.83084809603543374774581528358, −1.16778353231208119146753746316, 0,
1.16778353231208119146753746316, 2.83084809603543374774581528358, 3.42335796159392902091704565607, 4.41408406214326022029159289132, 5.41160569603738275873205852152, 6.78610692388255677658014028333, 7.58358075550629032868440981878, 8.151907807435224999155651612292, 9.032651152314828871476961988173
