| L(s) = 1 | + 14·3-s + 56·5-s − 49·7-s − 47·9-s − 232·11-s + 140·13-s + 784·15-s − 1.72e3·17-s + 98·19-s − 686·21-s + 1.82e3·23-s + 11·25-s − 4.06e3·27-s − 3.41e3·29-s − 7.64e3·31-s − 3.24e3·33-s − 2.74e3·35-s + 1.03e4·37-s + 1.96e3·39-s − 1.79e4·41-s − 1.08e4·43-s − 2.63e3·45-s + 9.32e3·47-s + 2.40e3·49-s − 2.41e4·51-s − 2.26e3·53-s − 1.29e4·55-s + ⋯ |
| L(s) = 1 | + 0.898·3-s + 1.00·5-s − 0.377·7-s − 0.193·9-s − 0.578·11-s + 0.229·13-s + 0.899·15-s − 1.44·17-s + 0.0622·19-s − 0.339·21-s + 0.718·23-s + 0.00351·25-s − 1.07·27-s − 0.754·29-s − 1.42·31-s − 0.519·33-s − 0.378·35-s + 1.24·37-s + 0.206·39-s − 1.66·41-s − 0.897·43-s − 0.193·45-s + 0.615·47-s + 1/7·49-s − 1.29·51-s − 0.110·53-s − 0.579·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 14 T + p^{5} T^{2} \) |
| 5 | \( 1 - 56 T + p^{5} T^{2} \) |
| 11 | \( 1 + 232 T + p^{5} T^{2} \) |
| 13 | \( 1 - 140 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1722 T + p^{5} T^{2} \) |
| 19 | \( 1 - 98 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1824 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3418 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7644 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10398 T + p^{5} T^{2} \) |
| 41 | \( 1 + 17962 T + p^{5} T^{2} \) |
| 43 | \( 1 + 10880 T + p^{5} T^{2} \) |
| 47 | \( 1 - 9324 T + p^{5} T^{2} \) |
| 53 | \( 1 + 2262 T + p^{5} T^{2} \) |
| 59 | \( 1 - 2730 T + p^{5} T^{2} \) |
| 61 | \( 1 + 25648 T + p^{5} T^{2} \) |
| 67 | \( 1 - 48404 T + p^{5} T^{2} \) |
| 71 | \( 1 + 58560 T + p^{5} T^{2} \) |
| 73 | \( 1 - 68082 T + p^{5} T^{2} \) |
| 79 | \( 1 - 31784 T + p^{5} T^{2} \) |
| 83 | \( 1 - 20538 T + p^{5} T^{2} \) |
| 89 | \( 1 + 50582 T + p^{5} T^{2} \) |
| 97 | \( 1 + 58506 T + p^{5} T^{2} \) |
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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
−9.572186423720103341902559475669, −9.072170088684689451020630962931, −8.198408723726811012819148359678, −7.07741871134170647338469129867, −6.06942450842665248693991303434, −5.12694795573011926895635483555, −3.69155321867730111082411823474, −2.61827535252308303707076241144, −1.82279737152664029030478630011, 0,
1.82279737152664029030478630011, 2.61827535252308303707076241144, 3.69155321867730111082411823474, 5.12694795573011926895635483555, 6.06942450842665248693991303434, 7.07741871134170647338469129867, 8.198408723726811012819148359678, 9.072170088684689451020630962931, 9.572186423720103341902559475669
