| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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
−12.06445015013949851453999474638, −11.24817683734094753711936799202, −9.966773655781626956546010508595, −8.483822123022907380484465042216, −8.131547140187430213274395118746, −6.77409075466514291700472522927, −4.91537525656401637745330793469, −3.62299926268921600296018605877, −2.27647681398025495199245603775, 0,
2.27647681398025495199245603775, 3.62299926268921600296018605877, 4.91537525656401637745330793469, 6.77409075466514291700472522927, 8.131547140187430213274395118746, 8.483822123022907380484465042216, 9.966773655781626956546010508595, 11.24817683734094753711936799202, 12.06445015013949851453999474638
