| L(s) = 1 | + (−5.16 − 0.546i)3-s + (5.21 + 1.89i)5-s + (1.58 − 9.00i)7-s + (26.4 + 5.64i)9-s + (−67.4 + 24.5i)11-s + (−63.1 + 53.0i)13-s + (−25.9 − 12.6i)15-s + (21.0 + 36.4i)17-s + (−22.5 + 39.0i)19-s + (−13.1 + 45.6i)21-s + (12.1 + 68.7i)23-s + (−72.1 − 60.5i)25-s + (−133. − 43.6i)27-s + (142. + 119. i)29-s + (−46.3 − 262. i)31-s + ⋯ |
| L(s) = 1 | + (−0.994 − 0.105i)3-s + (0.466 + 0.169i)5-s + (0.0857 − 0.486i)7-s + (0.977 + 0.209i)9-s + (−1.85 + 0.673i)11-s + (−1.34 + 1.13i)13-s + (−0.446 − 0.217i)15-s + (0.299 + 0.519i)17-s + (−0.271 + 0.471i)19-s + (−0.136 + 0.474i)21-s + (0.109 + 0.623i)23-s + (−0.577 − 0.484i)25-s + (−0.950 − 0.310i)27-s + (0.913 + 0.766i)29-s + (−0.268 − 1.52i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.150393 + 0.397588i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.150393 + 0.397588i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (5.16 + 0.546i)T \) |
| good | 5 | \( 1 + (-5.21 - 1.89i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-1.58 + 9.00i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (67.4 - 24.5i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (63.1 - 53.0i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-21.0 - 36.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (22.5 - 39.0i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-12.1 - 68.7i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-142. - 119. i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (46.3 + 262. i)T + (-2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (40.4 + 70.0i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (216. - 181. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (249. - 90.9i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (10.5 - 59.8i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 - 206.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-558. - 203. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-124. + 706. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-23.6 + 19.8i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-155. - 269. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (263. - 457. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (500. + 420. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (883. + 741. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-240. + 417. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (858. - 312. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
| 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
−13.40243344605391017008047105834, −12.57295666383077670650599147782, −11.55124059290266920256655440966, −10.27539947855210021707515701048, −9.895487135338227985551703588821, −7.82619082226860449591613412192, −6.91256674435146056007962899163, −5.53967783937979136940461276284, −4.49896453233113016287626100810, −2.08794427169218524383836782808,
0.24624619207289681444294600953, 2.67364281647306529076441056169, 5.11715437775329451297434125938, 5.49023287230430565373365026912, 7.11744069903886496203703948646, 8.386049634024690926232484495888, 10.02042180938651256774411487687, 10.51979840361603379638944018578, 11.85994670481888743905883491906, 12.72903787986577082673610373721
