| L(s) = 1 | + 357. i·3-s − 1.63e3·5-s − 3.37e3i·7-s − 6.85e4·9-s − 2.21e5i·11-s + 3.74e5·13-s − 5.82e5i·15-s − 4.30e5·17-s + 3.03e5i·19-s + 1.20e6·21-s − 8.29e6i·23-s − 7.10e6·25-s − 3.38e6i·27-s + 3.72e7·29-s + 7.54e6i·31-s + ⋯ |
| L(s) = 1 | + 1.46i·3-s − 0.521·5-s − 0.201i·7-s − 1.16·9-s − 1.37i·11-s + 1.00·13-s − 0.767i·15-s − 0.303·17-s + 0.122i·19-s + 0.295·21-s − 1.28i·23-s − 0.727·25-s − 0.236i·27-s + 1.81·29-s + 0.263i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.8257170059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8257170059\) |
| \(L(6)\) |
|
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 | \( 1 - 357. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 1.63e3T + 9.76e6T^{2} \) |
| 7 | \( 1 + 3.37e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 2.21e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 3.74e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 4.30e5T + 2.01e12T^{2} \) |
| 19 | \( 1 - 3.03e5iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 8.29e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 3.72e7T + 4.20e14T^{2} \) |
| 31 | \( 1 - 7.54e6iT - 8.19e14T^{2} \) |
| 37 | \( 1 + 7.50e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + 6.61e7T + 1.34e16T^{2} \) |
| 43 | \( 1 + 2.22e7iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 1.88e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 5.80e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 3.07e7iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 5.40e8T + 7.13e17T^{2} \) |
| 67 | \( 1 - 2.14e9iT - 1.82e18T^{2} \) |
| 71 | \( 1 - 2.89e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 3.51e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 4.62e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + 2.00e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 5.96e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 5.56e9T + 7.37e19T^{2} \) |
| 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
−10.75847068563615133131750499708, −9.919585797537829238346019635067, −8.667300067029334028796000342046, −8.338889200290990963796934050112, −6.64585023931264380380322289011, −5.61462804332597495728154947752, −4.47498321553629530014787296871, −3.74870209639840559058934719815, −2.89794042262585168465617657537, −0.979958002168731972175685970827,
0.18109411078936567331558215375, 1.38765635541593336990968138077, 2.09322740663015523082322455380, 3.45371078378069703056260598335, 4.76315671850284823871476581234, 6.11315300952420472411384180279, 6.93621557470175437666483558052, 7.70655996802814644417568132624, 8.494223721667508096450324195984, 9.683569490311432901119221969552
