| L(s) = 1 | − 614·5-s + 2.18e3·7-s − 4.94e3·11-s + 6.99e4·13-s − 3.76e5·17-s + 7.80e5·19-s − 1.76e6·23-s − 1.57e6·25-s + 3.21e6·29-s − 3.42e5·31-s − 1.34e6·35-s − 1.97e7·37-s + 1.58e7·41-s − 2.25e7·43-s − 4.89e7·47-s − 3.55e7·49-s − 5.23e7·53-s + 3.03e6·55-s − 7.70e7·59-s + 1.30e7·61-s − 4.29e7·65-s − 2.80e8·67-s + 8.85e7·71-s − 5.91e7·73-s − 1.07e7·77-s − 4.15e8·79-s − 4.28e7·83-s + ⋯ |
| L(s) = 1 | − 0.439·5-s + 0.343·7-s − 0.101·11-s + 0.679·13-s − 1.09·17-s + 1.37·19-s − 1.31·23-s − 0.806·25-s + 0.843·29-s − 0.0666·31-s − 0.151·35-s − 1.73·37-s + 0.877·41-s − 1.00·43-s − 1.46·47-s − 0.881·49-s − 0.911·53-s + 0.0446·55-s − 0.827·59-s + 0.120·61-s − 0.298·65-s − 1.70·67-s + 0.413·71-s − 0.243·73-s − 0.0349·77-s − 1.19·79-s − 0.0990·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 614 T + p^{9} T^{2} \) |
| 7 | \( 1 - 312 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 4940 T + p^{9} T^{2} \) |
| 13 | \( 1 - 69934 T + p^{9} T^{2} \) |
| 17 | \( 1 + 376978 T + p^{9} T^{2} \) |
| 19 | \( 1 - 780884 T + p^{9} T^{2} \) |
| 23 | \( 1 + 1764632 T + p^{9} T^{2} \) |
| 29 | \( 1 - 3212226 T + p^{9} T^{2} \) |
| 31 | \( 1 + 342880 T + p^{9} T^{2} \) |
| 37 | \( 1 + 19744506 T + p^{9} T^{2} \) |
| 41 | \( 1 - 15882390 T + p^{9} T^{2} \) |
| 43 | \( 1 + 22575764 T + p^{9} T^{2} \) |
| 47 | \( 1 + 48948528 T + p^{9} T^{2} \) |
| 53 | \( 1 + 52342550 T + p^{9} T^{2} \) |
| 59 | \( 1 + 77057660 T + p^{9} T^{2} \) |
| 61 | \( 1 - 13045726 T + p^{9} T^{2} \) |
| 67 | \( 1 + 280727164 T + p^{9} T^{2} \) |
| 71 | \( 1 - 88554680 T + p^{9} T^{2} \) |
| 73 | \( 1 + 59105654 T + p^{9} T^{2} \) |
| 79 | \( 1 + 415337264 T + p^{9} T^{2} \) |
| 83 | \( 1 + 42806932 T + p^{9} T^{2} \) |
| 89 | \( 1 - 803465958 T + p^{9} T^{2} \) |
| 97 | \( 1 - 674417762 T + p^{9} T^{2} \) |
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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
−12.02979837407677595540988841297, −11.23521806042375680411932133993, −9.977637131589480277530828042338, −8.633490327776755124628479285617, −7.61656661174934086651817039173, −6.22133898273995885426529227988, −4.75046761223572264230128470590, −3.40996752252021557429005444538, −1.67614993162122428515879650475, 0,
1.67614993162122428515879650475, 3.40996752252021557429005444538, 4.75046761223572264230128470590, 6.22133898273995885426529227988, 7.61656661174934086651817039173, 8.633490327776755124628479285617, 9.977637131589480277530828042338, 11.23521806042375680411932133993, 12.02979837407677595540988841297
