| L(s) = 1 | + 44·3-s + 430·5-s − 1.22e3·7-s − 251·9-s − 3.16e3·11-s + 6.11e3·13-s + 1.89e4·15-s − 1.62e4·17-s − 5.47e3·19-s − 5.38e4·21-s + 1.57e3·23-s + 1.06e5·25-s − 1.07e5·27-s + 1.22e5·29-s + 2.51e5·31-s − 1.39e5·33-s − 5.26e5·35-s − 5.23e4·37-s + 2.69e5·39-s − 3.19e5·41-s + 7.10e5·43-s − 1.07e5·45-s + 2.84e5·47-s + 6.74e5·49-s − 7.15e5·51-s + 2.96e5·53-s − 1.36e6·55-s + ⋯ |
| L(s) = 1 | + 0.940·3-s + 1.53·5-s − 1.34·7-s − 0.114·9-s − 0.716·11-s + 0.772·13-s + 1.44·15-s − 0.803·17-s − 0.183·19-s − 1.26·21-s + 0.0270·23-s + 1.36·25-s − 1.04·27-s + 0.935·29-s + 1.51·31-s − 0.674·33-s − 2.07·35-s − 0.169·37-s + 0.726·39-s − 0.723·41-s + 1.36·43-s − 0.176·45-s + 0.399·47-s + 0.819·49-s − 0.755·51-s + 0.273·53-s − 1.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.746102643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.746102643\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 44 T + p^{7} T^{2} \) |
| 5 | \( 1 - 86 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1224 T + p^{7} T^{2} \) |
| 11 | \( 1 + 3164 T + p^{7} T^{2} \) |
| 13 | \( 1 - 6118 T + p^{7} T^{2} \) |
| 17 | \( 1 + 16270 T + p^{7} T^{2} \) |
| 19 | \( 1 + 5476 T + p^{7} T^{2} \) |
| 23 | \( 1 - 1576 T + p^{7} T^{2} \) |
| 29 | \( 1 - 122838 T + p^{7} T^{2} \) |
| 31 | \( 1 - 251360 T + p^{7} T^{2} \) |
| 37 | \( 1 + 52338 T + p^{7} T^{2} \) |
| 41 | \( 1 + 319398 T + p^{7} T^{2} \) |
| 43 | \( 1 - 710788 T + p^{7} T^{2} \) |
| 47 | \( 1 - 284112 T + p^{7} T^{2} \) |
| 53 | \( 1 - 296062 T + p^{7} T^{2} \) |
| 59 | \( 1 + 897548 T + p^{7} T^{2} \) |
| 61 | \( 1 + 884810 T + p^{7} T^{2} \) |
| 67 | \( 1 - 4659692 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2710792 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5670854 T + p^{7} T^{2} \) |
| 79 | \( 1 + 5124176 T + p^{7} T^{2} \) |
| 83 | \( 1 + 1563556 T + p^{7} T^{2} \) |
| 89 | \( 1 - 11605674 T + p^{7} T^{2} \) |
| 97 | \( 1 - 10931618 T + p^{7} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.37732961213553518864318332349, −18.98264668318204185111092049867, −17.46648782044049269671089189901, −15.74973772472890962407064216729, −13.88085156374019789217664869614, −13.12724325687118163611734601502, −10.16184545039589024737271773228, −8.876344777673518550710912035257, −6.20294643883281071927713586447, −2.69066190630163132601067765020,
2.69066190630163132601067765020, 6.20294643883281071927713586447, 8.876344777673518550710912035257, 10.16184545039589024737271773228, 13.12724325687118163611734601502, 13.88085156374019789217664869614, 15.74973772472890962407064216729, 17.46648782044049269671089189901, 18.98264668318204185111092049867, 20.37732961213553518864318332349
