| L(s) = 1 | + (−1.24 + 0.671i)2-s + (−1.27 + 1.17i)3-s + (1.09 − 1.67i)4-s + (−2.10 − 0.759i)5-s + (0.795 − 2.31i)6-s − 0.738i·7-s + (−0.244 + 2.81i)8-s + (0.241 − 2.99i)9-s + (3.12 − 0.467i)10-s + (1.63 − 1.18i)11-s + (0.565 + 3.41i)12-s + (3.83 + 2.78i)13-s + (0.496 + 0.919i)14-s + (3.56 − 1.50i)15-s + (−1.58 − 3.67i)16-s + (3.35 − 1.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.880 + 0.474i)2-s + (−0.735 + 0.678i)3-s + (0.549 − 0.835i)4-s + (−0.940 − 0.339i)5-s + (0.324 − 0.945i)6-s − 0.279i·7-s + (−0.0863 + 0.996i)8-s + (0.0804 − 0.996i)9-s + (0.989 − 0.147i)10-s + (0.491 − 0.357i)11-s + (0.163 + 0.986i)12-s + (1.06 + 0.773i)13-s + (0.132 + 0.245i)14-s + (0.921 − 0.388i)15-s + (−0.397 − 0.917i)16-s + (0.813 − 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.581000 + 0.160594i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.581000 + 0.160594i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.24 - 0.671i)T \) |
| 3 | \( 1 + (1.27 - 1.17i)T \) |
| 5 | \( 1 + (2.10 + 0.759i)T \) |
| good | 7 | \( 1 + 0.738iT - 7T^{2} \) |
| 11 | \( 1 + (-1.63 + 1.18i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.83 - 2.78i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.35 + 1.08i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.82 - 0.919i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (1.01 - 0.736i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-7.24 - 2.35i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.78 + 2.20i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.994 + 0.722i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.02 + 1.40i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 + (-3.13 + 9.65i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (11.6 + 3.80i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.97 - 2.88i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.09 + 3.70i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-8.81 + 2.86i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-4.32 + 13.3i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (4.33 - 3.14i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (14.7 + 4.79i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.885 - 2.72i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.36 - 10.1i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.66 + 8.19i)T + (-78.4 - 57.0i)T^{2} \) |
| 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
−11.57145903735222924851961124213, −10.84675188950625435670922660724, −9.952896336871596583413217183209, −8.901334933045232574046010256007, −8.207415229773297198874730494930, −6.89437439792187306313526938838, −6.09232156455739943191430355706, −4.80271331926732175381617326926, −3.67483141083897030999775333535, −0.930660204043291772208105537316,
1.03069870839762905034799552353, 2.83514770340303893099853925359, 4.27480273198668689404724025148, 6.08717520032242260844637931191, 6.92400804099991850441899308979, 8.007188611138444308457437827555, 8.510345023585251040618279395842, 10.10518358014793188525693949870, 10.80457736444563628486124203177, 11.63301342272213425067728995938
