| L(s) = 1 | + 3.41·3-s − 0.558i·7-s + 2.66·9-s + (2.82 − 10.6i)11-s − 18.1i·13-s − 32.0i·17-s + 15.7i·19-s − 1.90i·21-s − 31.7·23-s − 21.6·27-s + 48.3i·29-s − 43.3·31-s + (9.63 − 36.3i)33-s − 21.4·37-s − 61.8i·39-s + ⋯ |
| L(s) = 1 | + 1.13·3-s − 0.0798i·7-s + 0.296·9-s + (0.256 − 0.966i)11-s − 1.39i·13-s − 1.88i·17-s + 0.828i·19-s − 0.0908i·21-s − 1.38·23-s − 0.801·27-s + 1.66i·29-s − 1.39·31-s + (0.291 − 1.10i)33-s − 0.579·37-s − 1.58i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.101984454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.101984454\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-2.82 + 10.6i)T \) |
| good | 3 | \( 1 - 3.41T + 9T^{2} \) |
| 7 | \( 1 + 0.558iT - 49T^{2} \) |
| 13 | \( 1 + 18.1iT - 169T^{2} \) |
| 17 | \( 1 + 32.0iT - 289T^{2} \) |
| 19 | \( 1 - 15.7iT - 361T^{2} \) |
| 23 | \( 1 + 31.7T + 529T^{2} \) |
| 29 | \( 1 - 48.3iT - 841T^{2} \) |
| 31 | \( 1 + 43.3T + 961T^{2} \) |
| 37 | \( 1 + 21.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 38.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 75.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 8.43T + 2.80e3T^{2} \) |
| 59 | \( 1 + 29.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 64.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 18.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 94.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 0.945iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 73.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 161. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 91.5T + 7.92e3T^{2} \) |
| 97 | \( 1 - 33.1T + 9.40e3T^{2} \) |
| 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
−9.118601144541406062123364267158, −8.756393943036475269173879836241, −7.76887281247716116082907075994, −7.28858885534177793606740217578, −5.88513750525548452442891740874, −5.22510329761939024673343104199, −3.67545400396807706891286091046, −3.21187951710206860130024400685, −2.10995734993204871729694823716, −0.50832468429743735143525845271,
1.79153770688133166670528255821, 2.37669566422374290195358269679, 3.91236070077916206465907960992, 4.22215916989879522826921432501, 5.76403489352851384325915939907, 6.65439771406830534507041116318, 7.59273968053250140447774575726, 8.310735605166026299840166279332, 9.111595426018866479424576894087, 9.622080100084117902951907364785
