L(s) = 1 | + 2.98i·2-s − 4.88·4-s + 7.85i·5-s + 2.80·7-s − 2.63i·8-s − 23.4·10-s − 1.51·13-s + 8.36i·14-s − 11.6·16-s − 26.3i·17-s − 14.5·19-s − 38.3i·20-s − 25.0i·23-s − 36.7·25-s − 4.52i·26-s + ⋯ |
L(s) = 1 | + 1.49i·2-s − 1.22·4-s + 1.57i·5-s + 0.401·7-s − 0.328i·8-s − 2.34·10-s − 0.116·13-s + 0.597i·14-s − 0.730·16-s − 1.55i·17-s − 0.765·19-s − 1.91i·20-s − 1.09i·23-s − 1.46·25-s − 0.174i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5261447421\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5261447421\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.98iT - 4T^{2} \) |
| 5 | \( 1 - 7.85iT - 25T^{2} \) |
| 7 | \( 1 - 2.80T + 49T^{2} \) |
| 13 | \( 1 + 1.51T + 169T^{2} \) |
| 17 | \( 1 + 26.3iT - 289T^{2} \) |
| 19 | \( 1 + 14.5T + 361T^{2} \) |
| 23 | \( 1 + 25.0iT - 529T^{2} \) |
| 29 | \( 1 - 50.7iT - 841T^{2} \) |
| 31 | \( 1 + 51.6T + 961T^{2} \) |
| 37 | \( 1 + 19.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 30.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 2.11T + 1.84e3T^{2} \) |
| 47 | \( 1 + 40.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 41.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 96.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 2.89T + 3.72e3T^{2} \) |
| 67 | \( 1 + 88.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 11.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 104.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 16.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 22.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 30.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 28.5T + 9.40e3T^{2} \) |
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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
−10.40398325616783559280845780673, −9.271042729592737230414518600166, −8.515884966033853460438639838517, −7.50546196352377333592120673121, −7.01719069380432835576963098609, −6.48926631695578568257349896837, −5.46427035528540241434709044746, −4.64885652684298364103010166288, −3.30291374627515643637271821948, −2.23448432197963852419589823106,
0.15437103302540633261621272540, 1.44718016771864347683766671812, 2.03826915021584699995434766366, 3.66218603322402702883610040437, 4.27319219939356709255451503233, 5.17707619319038258989353658285, 6.15616377459224818846736595141, 7.69273969286406451726933808239, 8.469616947231964482382629205118, 9.180349153888961759504566707921