| L(s) = 1 | + (1.89 + 0.627i)2-s + (3.21 + 2.38i)4-s + (−1.53 + 2.65i)5-s + (−0.720 − 1.24i)7-s + (4.60 + 6.54i)8-s + (−4.57 + 4.07i)10-s + (8.82 + 15.2i)11-s + (8.00 + 4.61i)13-s + (−0.584 − 2.82i)14-s + (4.63 + 15.3i)16-s − 4.69i·17-s − 16.8i·19-s + (−11.2 + 4.86i)20-s + (7.16 + 34.5i)22-s + (−33.8 − 19.5i)23-s + ⋯ |
| L(s) = 1 | + (0.949 + 0.313i)2-s + (0.803 + 0.595i)4-s + (−0.306 + 0.530i)5-s + (−0.102 − 0.178i)7-s + (0.575 + 0.817i)8-s + (−0.457 + 0.407i)10-s + (0.802 + 1.39i)11-s + (0.615 + 0.355i)13-s + (−0.0417 − 0.201i)14-s + (0.289 + 0.957i)16-s − 0.276i·17-s − 0.888i·19-s + (−0.562 + 0.243i)20-s + (0.325 + 1.57i)22-s + (−1.47 − 0.849i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.18318 + 1.48940i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.18318 + 1.48940i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.89 - 0.627i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.53 - 2.65i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (0.720 + 1.24i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.82 - 15.2i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-8.00 - 4.61i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 4.69iT - 289T^{2} \) |
| 19 | \( 1 + 16.8iT - 361T^{2} \) |
| 23 | \( 1 + (33.8 + 19.5i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (7.60 + 13.1i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-24.3 + 42.1i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 14.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (8.78 + 5.07i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-19.3 + 11.1i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-36.8 + 21.2i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 71.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-34.9 + 60.6i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (89.7 - 51.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (10.5 + 6.07i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 112. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 84.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (22.9 + 39.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-25.7 - 44.6i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 105. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-19.4 - 33.6i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.22822309176918487303001675519, −11.60836351895290461826237819140, −10.58667107484459197412421800616, −9.361359855661667238009031293587, −7.920366125916120210513688023448, −6.96439521575649948966015796527, −6.23797352316293078755843010685, −4.64861246033288799941816606246, −3.79407012445157485375509305165, −2.22606562656401599882924181383,
1.27597082791384709701065207625, 3.25100212873335151125991411640, 4.18516759643362194952628250309, 5.66056070403072540746914082678, 6.32058946375472848876446657355, 7.907010318776568569284959894699, 8.913526462937473925350096406281, 10.25191559118782246065280066854, 11.18550796047554313796253109759, 12.06356611166346595504334476519
