| L(s) = 1 | + (−1.29 + 0.563i)2-s + (0.654 + 0.755i)3-s + (1.36 − 1.46i)4-s + (0.355 + 2.46i)5-s + (−1.27 − 0.611i)6-s + (−0.661 + 1.44i)7-s + (−0.949 + 2.66i)8-s + (−0.142 + 0.989i)9-s + (−1.85 − 3.00i)10-s + (−0.328 + 1.12i)11-s + (1.99 + 0.0756i)12-s + (−3.21 + 1.47i)13-s + (0.0425 − 2.25i)14-s + (−1.63 + 1.88i)15-s + (−0.268 − 3.99i)16-s + (−0.248 + 0.387i)17-s + ⋯ |
| L(s) = 1 | + (−0.917 + 0.398i)2-s + (0.378 + 0.436i)3-s + (0.682 − 0.730i)4-s + (0.158 + 1.10i)5-s + (−0.520 − 0.249i)6-s + (−0.250 + 0.547i)7-s + (−0.335 + 0.941i)8-s + (−0.0474 + 0.329i)9-s + (−0.585 − 0.949i)10-s + (−0.0991 + 0.337i)11-s + (0.576 + 0.0218i)12-s + (−0.892 + 0.407i)13-s + (0.0113 − 0.601i)14-s + (−0.421 + 0.486i)15-s + (−0.0671 − 0.997i)16-s + (−0.0603 + 0.0939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.123582 + 0.794980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.123582 + 0.794980i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.29 - 0.563i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-4.16 - 2.38i)T \) |
| good | 5 | \( 1 + (-0.355 - 2.46i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (0.661 - 1.44i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.328 - 1.12i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (3.21 - 1.47i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.248 - 0.387i)T + (-7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (3.28 + 5.10i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (3.40 - 5.30i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (0.449 + 0.389i)T + (4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.0548 + 0.381i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.738 + 5.13i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-2.16 + 1.87i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 0.152iT - 47T^{2} \) |
| 53 | \( 1 + (3.00 - 6.59i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-0.844 - 1.85i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (6.17 - 7.12i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.430 - 1.46i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (0.422 + 1.43i)T + (-59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (9.07 - 5.83i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (3.40 + 7.46i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-4.37 - 0.628i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (2.04 - 1.76i)T + (12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-10.0 + 1.44i)T + (93.0 - 27.3i)T^{2} \) |
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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
−10.80592810078199090268876228539, −10.27624717531497431346108094689, −9.213590445921044397082305353082, −8.889042189273319945830847757019, −7.40631500926058417608859381581, −7.00721629689404063308038074167, −5.90909061716439248360658791583, −4.76523769625123188321762129466, −3.00587940599875890287963000109, −2.19538450345481828007184337223,
0.56807884718624310185765350931, 1.91311062507312714909324916989, 3.25260419494962799307621131511, 4.53345127301686671133941735744, 5.97666367137433471605086367370, 7.09037950890065844595462293162, 7.974780110466824330253293374060, 8.589908207089302198238647459455, 9.497603501585921312536556905574, 10.16422314364220358095496095094
