| 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
−9.913071822640726152024743688614, −9.056049294397904300488206588404, −8.093557315912239316759332245286, −6.61245109070683883247631332219, −6.14861202931696972424440068660, −5.19502119362275894424827926879, −4.19023577693847241362096888082, −2.51011321749267820896969986293, −1.39771872769074897531002236466, 0,
1.39771872769074897531002236466, 2.51011321749267820896969986293, 4.19023577693847241362096888082, 5.19502119362275894424827926879, 6.14861202931696972424440068660, 6.61245109070683883247631332219, 8.093557315912239316759332245286, 9.056049294397904300488206588404, 9.913071822640726152024743688614
