| L(s) = 1 | + 16·2-s + 34·3-s + 256·4-s + 544·6-s + 4.09e3·8-s − 5.40e3·9-s − 2.71e4·11-s + 8.70e3·12-s + 6.55e4·16-s + 1.62e5·17-s − 8.64e4·18-s − 7.22e4·19-s − 4.34e5·22-s + 1.39e5·24-s + 3.90e5·25-s − 4.06e5·27-s + 1.04e6·32-s − 9.23e5·33-s + 2.59e6·34-s − 1.38e6·36-s − 1.15e6·38-s − 4.09e6·41-s + 5.42e6·43-s − 6.95e6·44-s + 2.22e6·48-s + 5.76e6·49-s + 6.25e6·50-s + ⋯ |
| L(s) = 1 | + 2-s + 0.419·3-s + 4-s + 0.419·6-s + 8-s − 0.823·9-s − 1.85·11-s + 0.419·12-s + 16-s + 1.94·17-s − 0.823·18-s − 0.554·19-s − 1.85·22-s + 0.419·24-s + 25-s − 0.765·27-s + 32-s − 0.778·33-s + 1.94·34-s − 0.823·36-s − 0.554·38-s − 1.45·41-s + 1.58·43-s − 1.85·44-s + 0.419·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.503624620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.503624620\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{4} T \) |
| good | 3 | \( 1 - 34 T + p^{8} T^{2} \) |
| 5 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 7 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 11 | \( 1 + 27166 T + p^{8} T^{2} \) |
| 13 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 17 | \( 1 - 162434 T + p^{8} T^{2} \) |
| 19 | \( 1 + 72286 T + p^{8} T^{2} \) |
| 23 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 37 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 41 | \( 1 + 4099006 T + p^{8} T^{2} \) |
| 43 | \( 1 - 5426402 T + p^{8} T^{2} \) |
| 47 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 53 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 59 | \( 1 + 24178078 T + p^{8} T^{2} \) |
| 61 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 67 | \( 1 + 13944286 T + p^{8} T^{2} \) |
| 71 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 73 | \( 1 - 33567554 T + p^{8} T^{2} \) |
| 79 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 83 | \( 1 - 30209954 T + p^{8} T^{2} \) |
| 89 | \( 1 + 95519806 T + p^{8} T^{2} \) |
| 97 | \( 1 + 77418238 T + p^{8} T^{2} \) |
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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
−20.43636867953846728421144808853, −18.83928285793122858494437083348, −16.66903314807526621007903369815, −15.17267667565937332290132996230, −13.90109340119322330440806244710, −12.46077624024266586220101207077, −10.60055291757551314740460033149, −7.86918561829260166901729442465, −5.45146516011363861796460456008, −2.90375573006020318424530649363,
2.90375573006020318424530649363, 5.45146516011363861796460456008, 7.86918561829260166901729442465, 10.60055291757551314740460033149, 12.46077624024266586220101207077, 13.90109340119322330440806244710, 15.17267667565937332290132996230, 16.66903314807526621007903369815, 18.83928285793122858494437083348, 20.43636867953846728421144808853
