| L(s) = 1 | + (0.900 − 0.433i)2-s + (−1.64 + 0.550i)3-s + (0.623 − 0.781i)4-s + (1.41 + 1.31i)5-s + (−1.24 + 1.20i)6-s + (2.64 − 0.0335i)7-s + (0.222 − 0.974i)8-s + (2.39 − 1.80i)9-s + (1.84 + 0.570i)10-s + (1.64 + 1.12i)11-s + (−0.593 + 1.62i)12-s + (1.72 + 1.17i)13-s + (2.36 − 1.17i)14-s + (−3.05 − 1.38i)15-s + (−0.222 − 0.974i)16-s + (−7.35 + 1.10i)17-s + ⋯ |
| L(s) = 1 | + (0.637 − 0.306i)2-s + (−0.948 + 0.317i)3-s + (0.311 − 0.390i)4-s + (0.634 + 0.588i)5-s + (−0.506 + 0.493i)6-s + (0.999 − 0.0126i)7-s + (0.0786 − 0.344i)8-s + (0.798 − 0.602i)9-s + (0.584 + 0.180i)10-s + (0.496 + 0.338i)11-s + (−0.171 + 0.469i)12-s + (0.478 + 0.326i)13-s + (0.633 − 0.314i)14-s + (−0.788 − 0.356i)15-s + (−0.0556 − 0.243i)16-s + (−1.78 + 0.268i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.13870 + 0.224236i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13870 + 0.224236i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.900 + 0.433i)T \) |
| 3 | \( 1 + (1.64 - 0.550i)T \) |
| 7 | \( 1 + (-2.64 + 0.0335i)T \) |
| good | 5 | \( 1 + (-1.41 - 1.31i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (-1.64 - 1.12i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (-1.72 - 1.17i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (7.35 - 1.10i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (2.17 + 3.77i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.74 - 7.00i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (-9.29 + 1.40i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 + (1.18 - 3.02i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-9.56 + 2.95i)T + (33.8 - 23.0i)T^{2} \) |
| 43 | \( 1 + (-11.9 - 3.68i)T + (35.5 + 24.2i)T^{2} \) |
| 47 | \( 1 + (4.78 - 2.30i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (3.49 + 8.89i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-1.03 - 4.53i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (1.19 + 1.50i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + (4.32 - 5.42i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (0.327 - 0.223i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + 8.80T + 79T^{2} \) |
| 83 | \( 1 + (11.8 - 8.09i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (-0.231 - 3.08i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (-4.51 + 7.82i)T + (-48.5 - 84.0i)T^{2} \) |
| 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
−10.49143362888717418842934152616, −9.558797840449893341171917512696, −8.688496747370850122286069239697, −7.17694503325888083906150475917, −6.49997749083609400149047077580, −5.81093414558611999843194073929, −4.61765601232365248335037434994, −4.26095140504974107579489993066, −2.58324279939424237883774048518, −1.40095261730649592750004311828,
1.15937658836639331426008695752, 2.38086360896408848320760446625, 4.36149852915047490268506130066, 4.72141471142062999643989702807, 5.85975723843965828189090238409, 6.32957629774886198359429829152, 7.33437938802988200563196029137, 8.422231704356213958922781836700, 9.029454615461137931316881507615, 10.56450330917823080183010082542
