| L(s) = 1 | − 5.28e3·5-s + 4.90e4·7-s + 4.14e5·11-s − 5.22e5·13-s − 9.49e6·17-s − 1.30e7·19-s + 5.87e7·23-s − 2.09e7·25-s + 1.17e8·29-s − 1.42e8·31-s − 2.58e8·35-s + 7.18e8·37-s + 6.68e8·41-s − 1.41e8·43-s + 7.29e8·47-s + 4.27e8·49-s − 4.91e9·53-s − 2.18e9·55-s + 1.40e9·59-s − 3.22e9·61-s + 2.76e9·65-s + 2.35e9·67-s − 2.22e10·71-s − 2.80e10·73-s + 2.03e10·77-s + 2.06e10·79-s + 3.78e10·83-s + ⋯ |
| L(s) = 1 | − 0.755·5-s + 1.10·7-s + 0.775·11-s − 0.390·13-s − 1.62·17-s − 1.20·19-s + 1.90·23-s − 0.429·25-s + 1.06·29-s − 0.896·31-s − 0.833·35-s + 1.70·37-s + 0.900·41-s − 0.146·43-s + 0.463·47-s + 0.216·49-s − 1.61·53-s − 0.586·55-s + 0.256·59-s − 0.488·61-s + 0.295·65-s + 0.213·67-s − 1.46·71-s − 1.58·73-s + 0.855·77-s + 0.756·79-s + 1.05·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 1056 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 49036 T + p^{11} T^{2} \) |
| 11 | \( 1 - 414336 T + p^{11} T^{2} \) |
| 13 | \( 1 + 522982 T + p^{11} T^{2} \) |
| 17 | \( 1 + 9499968 T + p^{11} T^{2} \) |
| 19 | \( 1 + 13053944 T + p^{11} T^{2} \) |
| 23 | \( 1 - 58755840 T + p^{11} T^{2} \) |
| 29 | \( 1 - 117142944 T + p^{11} T^{2} \) |
| 31 | \( 1 + 142907156 T + p^{11} T^{2} \) |
| 37 | \( 1 - 718521806 T + p^{11} T^{2} \) |
| 41 | \( 1 - 668055360 T + p^{11} T^{2} \) |
| 43 | \( 1 + 141575864 T + p^{11} T^{2} \) |
| 47 | \( 1 - 729235200 T + p^{11} T^{2} \) |
| 53 | \( 1 + 4917225312 T + p^{11} T^{2} \) |
| 59 | \( 1 - 1408015104 T + p^{11} T^{2} \) |
| 61 | \( 1 + 3223327018 T + p^{11} T^{2} \) |
| 67 | \( 1 - 2358681328 T + p^{11} T^{2} \) |
| 71 | \( 1 + 22245092352 T + p^{11} T^{2} \) |
| 73 | \( 1 + 28036594330 T + p^{11} T^{2} \) |
| 79 | \( 1 - 20685045676 T + p^{11} T^{2} \) |
| 83 | \( 1 - 37818604416 T + p^{11} T^{2} \) |
| 89 | \( 1 + 11288711808 T + p^{11} T^{2} \) |
| 97 | \( 1 + 115724393266 T + p^{11} T^{2} \) |
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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
−10.95102812131898964173829480273, −9.251946191746197987693849793204, −8.446670617667862036979855334369, −7.41347227021840008363247522628, −6.37830403173297350533803035818, −4.75529651656870995191578033026, −4.16783942913277502713281959460, −2.56418734419095246934277629102, −1.31990563405057561134582516612, 0,
1.31990563405057561134582516612, 2.56418734419095246934277629102, 4.16783942913277502713281959460, 4.75529651656870995191578033026, 6.37830403173297350533803035818, 7.41347227021840008363247522628, 8.446670617667862036979855334369, 9.251946191746197987693849793204, 10.95102812131898964173829480273
