| L(s) = 1 | − 0.228·3-s + 5-s − 3.99i·7-s − 2.94·9-s − 3.75·11-s − 3.12i·13-s − 0.228·15-s + (3.25 + 2.53i)17-s − 1.96i·19-s + 0.913i·21-s + 0.990i·23-s + 25-s + 1.36·27-s − 4.34·29-s + 2.33i·31-s + ⋯ |
| L(s) = 1 | − 0.132·3-s + 0.447·5-s − 1.50i·7-s − 0.982·9-s − 1.13·11-s − 0.865i·13-s − 0.0590·15-s + (0.788 + 0.614i)17-s − 0.450i·19-s + 0.199i·21-s + 0.206i·23-s + 0.200·25-s + 0.261·27-s − 0.806·29-s + 0.418i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2105202527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2105202527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 + (-3.25 - 2.53i)T \) |
| good | 3 | \( 1 + 0.228T + 3T^{2} \) |
| 7 | \( 1 + 3.99iT - 7T^{2} \) |
| 11 | \( 1 + 3.75T + 11T^{2} \) |
| 13 | \( 1 + 3.12iT - 13T^{2} \) |
| 19 | \( 1 + 1.96iT - 19T^{2} \) |
| 23 | \( 1 - 0.990iT - 23T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 - 2.33iT - 31T^{2} \) |
| 37 | \( 1 + 6.42T + 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 3.10iT - 43T^{2} \) |
| 47 | \( 1 - 0.730T + 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 14.8iT - 59T^{2} \) |
| 61 | \( 1 + 8.99T + 61T^{2} \) |
| 67 | \( 1 - 7.33iT - 67T^{2} \) |
| 71 | \( 1 + 0.713iT - 71T^{2} \) |
| 73 | \( 1 - 7.10iT - 73T^{2} \) |
| 79 | \( 1 + 8.64iT - 79T^{2} \) |
| 83 | \( 1 + 11.2iT - 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 18.7iT - 97T^{2} \) |
| 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
−8.295815272799353033457780899433, −7.59375571739973540929267639646, −7.05820357037991075769452759368, −5.83288611216585781255547441882, −5.49319731264379180520648764126, −4.47516921878598492022600790706, −3.43175605354755679958937188824, −2.71555514081169997087213981419, −1.28776227588187789613747834212, −0.06749007609547348601241389146,
1.88997022360410384568122669388, 2.60471351825840699393896929727, 3.42945798777566600301144200710, 4.95169402129345325087160534369, 5.42292388427522692562226933289, 5.99722148020157010674337280187, 6.86209412489269046608888431350, 8.010988839508126457821085470893, 8.438872631710661639872960923817, 9.361608342270591474760779528783
