| L(s) = 1 | + (−0.104 − 0.994i)2-s + (1.50 + 1.66i)3-s + (−0.978 + 0.207i)4-s + (2.11 + 0.739i)5-s + (1.50 − 1.66i)6-s + 0.927·7-s + (0.309 + 0.951i)8-s + (−0.213 + 2.03i)9-s + (0.515 − 2.17i)10-s + (−2.60 + 1.89i)11-s + (−1.81 − 1.31i)12-s + (−0.121 + 1.15i)13-s + (−0.0969 − 0.921i)14-s + (1.93 + 4.63i)15-s + (0.913 − 0.406i)16-s + (5.51 + 1.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (0.867 + 0.963i)3-s + (−0.489 + 0.103i)4-s + (0.943 + 0.330i)5-s + (0.613 − 0.681i)6-s + 0.350·7-s + (0.109 + 0.336i)8-s + (−0.0711 + 0.677i)9-s + (0.162 − 0.688i)10-s + (−0.786 + 0.571i)11-s + (−0.524 − 0.381i)12-s + (−0.0335 + 0.319i)13-s + (−0.0258 − 0.246i)14-s + (0.499 + 1.19i)15-s + (0.228 − 0.101i)16-s + (1.33 + 0.284i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.24191 + 0.601767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.24191 + 0.601767i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 5 | \( 1 + (-2.11 - 0.739i)T \) |
| 19 | \( 1 + (1.43 + 4.11i)T \) |
| good | 3 | \( 1 + (-1.50 - 1.66i)T + (-0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 - 0.927T + 7T^{2} \) |
| 11 | \( 1 + (2.60 - 1.89i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.121 - 1.15i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-5.51 - 1.17i)T + (15.5 + 6.91i)T^{2} \) |
| 23 | \( 1 + (-4.54 - 2.02i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (3.42 - 0.727i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (-1.18 - 3.63i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (7.35 + 5.34i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.0589 + 0.0262i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (6.06 - 10.4i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.54 - 0.966i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-9.68 + 2.05i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-1.51 + 0.673i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-8.04 - 3.58i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-7.16 + 7.96i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (7.17 + 7.96i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (1.01 + 9.68i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (1.72 + 1.91i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-2.00 - 6.15i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (2.98 + 1.33i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (-7.24 - 8.04i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−10.08733749484441267340646908158, −9.433590660935350241514024404652, −8.819901094987566791664910150670, −7.86522154234054832402790533809, −6.79039050311874448269705578966, −5.34636735087703133593389980493, −4.77237983842083157918724813564, −3.49844680885018384245400026911, −2.78650514240804262763139587871, −1.69251001903257822733379847121,
1.13350413904644938722498450128, 2.30649680539510851535593270958, 3.43325585753345269664946610996, 5.12591174932091862365881250922, 5.62608722857996780442035942203, 6.72987398921225100564647120303, 7.53160534056883479480592246825, 8.370692007598893555044537273532, 8.643562341005047146145095566474, 9.896269412152425097605584724643
