| L(s) = 1 | − 105·5-s + 937·7-s + 5.94e3·11-s + 68·13-s + 5.40e3·17-s + 4.83e4·19-s − 642·23-s − 6.71e4·25-s + 1.25e5·29-s + 1.61e5·31-s − 9.83e4·35-s − 4.14e5·37-s + 6.27e5·41-s − 5.70e5·43-s + 5.38e5·47-s + 5.44e4·49-s − 3.56e5·53-s − 6.24e5·55-s − 2.91e6·59-s + 2.68e6·61-s − 7.14e3·65-s − 2.68e6·67-s − 3.70e6·71-s − 1.53e5·73-s + 5.56e6·77-s + 7.57e6·79-s + 9.34e6·83-s + ⋯ |
| L(s) = 1 | − 0.375·5-s + 1.03·7-s + 1.34·11-s + 0.00858·13-s + 0.266·17-s + 1.61·19-s − 0.0110·23-s − 0.858·25-s + 0.958·29-s + 0.972·31-s − 0.387·35-s − 1.34·37-s + 1.42·41-s − 1.09·43-s + 0.756·47-s + 0.0660·49-s − 0.328·53-s − 0.505·55-s − 1.84·59-s + 1.51·61-s − 0.00322·65-s − 1.08·67-s − 1.22·71-s − 0.0460·73-s + 1.39·77-s + 1.72·79-s + 1.79·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.961445854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.961445854\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 21 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 937 T + p^{7} T^{2} \) |
| 11 | \( 1 - 5943 T + p^{7} T^{2} \) |
| 13 | \( 1 - 68 T + p^{7} T^{2} \) |
| 17 | \( 1 - 5400 T + p^{7} T^{2} \) |
| 19 | \( 1 - 48382 T + p^{7} T^{2} \) |
| 23 | \( 1 + 642 T + p^{7} T^{2} \) |
| 29 | \( 1 - 125934 T + p^{7} T^{2} \) |
| 31 | \( 1 - 161275 T + p^{7} T^{2} \) |
| 37 | \( 1 + 414286 T + p^{7} T^{2} \) |
| 41 | \( 1 - 627474 T + p^{7} T^{2} \) |
| 43 | \( 1 + 570590 T + p^{7} T^{2} \) |
| 47 | \( 1 - 538698 T + p^{7} T^{2} \) |
| 53 | \( 1 + 356283 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2910828 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2684168 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2681078 T + p^{7} T^{2} \) |
| 71 | \( 1 + 3705480 T + p^{7} T^{2} \) |
| 73 | \( 1 + 153151 T + p^{7} T^{2} \) |
| 79 | \( 1 - 7579288 T + p^{7} T^{2} \) |
| 83 | \( 1 - 9345999 T + p^{7} T^{2} \) |
| 89 | \( 1 + 4033602 T + p^{7} T^{2} \) |
| 97 | \( 1 + 5754097 T + p^{7} T^{2} \) |
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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
−9.932817829714748790656927936889, −9.051840536998555147882261269511, −8.106327500287249488381870303199, −7.35594774069496508366788585913, −6.27347982994394743555017995691, −5.13460555103924950313853549168, −4.21611048926358655399833841721, −3.18411894401436961423241383627, −1.68175389370521877062059606646, −0.847273473448677422215821589910,
0.847273473448677422215821589910, 1.68175389370521877062059606646, 3.18411894401436961423241383627, 4.21611048926358655399833841721, 5.13460555103924950313853549168, 6.27347982994394743555017995691, 7.35594774069496508366788585913, 8.106327500287249488381870303199, 9.051840536998555147882261269511, 9.932817829714748790656927936889
