| L(s) = 1 | + (0.888 − 0.458i)2-s + (−0.786 + 0.618i)3-s + (0.580 − 0.814i)4-s + (−1.51 + 1.44i)5-s + (−0.415 + 0.909i)6-s + (−0.846 − 2.50i)7-s + (0.142 − 0.989i)8-s + (0.235 − 0.971i)9-s + (−0.686 + 1.98i)10-s + (0.565 + 0.291i)11-s + (0.0475 + 0.998i)12-s + (2.58 + 2.97i)13-s + (−1.90 − 1.83i)14-s + (0.298 − 2.07i)15-s + (−0.327 − 0.945i)16-s + (7.32 − 0.699i)17-s + ⋯ |
| L(s) = 1 | + (0.628 − 0.324i)2-s + (−0.453 + 0.356i)3-s + (0.290 − 0.407i)4-s + (−0.679 + 0.647i)5-s + (−0.169 + 0.371i)6-s + (−0.320 − 0.947i)7-s + (0.0503 − 0.349i)8-s + (0.0785 − 0.323i)9-s + (−0.217 + 0.627i)10-s + (0.170 + 0.0878i)11-s + (0.0137 + 0.288i)12-s + (0.715 + 0.826i)13-s + (−0.508 − 0.491i)14-s + (0.0771 − 0.536i)15-s + (−0.0817 − 0.236i)16-s + (1.77 − 0.169i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.72520 - 0.449099i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.72520 - 0.449099i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.888 + 0.458i)T \) |
| 3 | \( 1 + (0.786 - 0.618i)T \) |
| 7 | \( 1 + (0.846 + 2.50i)T \) |
| 23 | \( 1 + (-0.166 + 4.79i)T \) |
| good | 5 | \( 1 + (1.51 - 1.44i)T + (0.237 - 4.99i)T^{2} \) |
| 11 | \( 1 + (-0.565 - 0.291i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (-2.58 - 2.97i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-7.32 + 0.699i)T + (16.6 - 3.21i)T^{2} \) |
| 19 | \( 1 + (1.42 + 0.135i)T + (18.6 + 3.59i)T^{2} \) |
| 29 | \( 1 + (-3.42 + 7.49i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-2.68 + 1.07i)T + (22.4 - 21.3i)T^{2} \) |
| 37 | \( 1 + (-0.689 + 2.84i)T + (-32.8 - 16.9i)T^{2} \) |
| 41 | \( 1 + (-7.09 - 2.08i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.439 - 3.05i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (-5.96 + 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.28 - 1.01i)T + (49.2 + 19.6i)T^{2} \) |
| 59 | \( 1 + (1.62 - 4.70i)T + (-46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (-6.27 - 4.93i)T + (14.3 + 59.2i)T^{2} \) |
| 67 | \( 1 + (0.284 - 5.97i)T + (-66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (0.235 + 0.151i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.82 + 2.55i)T + (-23.8 - 68.9i)T^{2} \) |
| 79 | \( 1 + (-4.06 + 0.783i)T + (73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (13.5 - 3.97i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-5.66 - 2.26i)T + (64.4 + 61.4i)T^{2} \) |
| 97 | \( 1 + (8.33 + 2.44i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.26081735003215245819787663131, −9.449440462873862128183362553105, −8.123025909403394854677994099352, −7.19150688333736198721490856395, −6.49475182234940341428269859998, −5.61199866255802437000314084226, −4.21378481511963790219125584578, −3.94150512753903258228711156695, −2.79484481680127783858018195829, −0.925446924247027647951091580812,
1.15628580840039172410819809896, 2.92318274996430597795570785672, 3.83768780161484716380240952927, 5.10843615589753502638613047533, 5.66816847421004567230862032971, 6.45413077992483203728482234197, 7.63742395712143193859869649903, 8.203950978816944335501698403156, 9.028994864803876322909098620068, 10.18853223152536272092341175975
