| L(s) = 1 | + (0.234 + 0.536i)2-s + (1.12 − 1.21i)4-s + (1.79 − 0.269i)5-s + (2.59 − 2.40i)7-s + (2.02 + 0.707i)8-s + (0.564 + 0.898i)10-s + (0.974 + 0.838i)11-s + (−2.06 − 3.02i)13-s + (1.89 + 0.828i)14-s + (−0.154 − 2.05i)16-s + (2.82 − 2.82i)17-s + (−5.81 + 3.65i)19-s + (1.69 − 2.48i)20-s + (−0.221 + 0.719i)22-s + (−8.33 + 3.27i)23-s + ⋯ |
| L(s) = 1 | + (0.165 + 0.379i)2-s + (0.563 − 0.607i)4-s + (0.801 − 0.120i)5-s + (0.980 − 0.909i)7-s + (0.714 + 0.250i)8-s + (0.178 + 0.283i)10-s + (0.293 + 0.252i)11-s + (−0.571 − 0.838i)13-s + (0.507 + 0.221i)14-s + (−0.0385 − 0.513i)16-s + (0.684 − 0.684i)17-s + (−1.33 + 0.838i)19-s + (0.378 − 0.554i)20-s + (−0.0472 + 0.153i)22-s + (−1.73 + 0.682i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.35598 - 0.534427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.35598 - 0.534427i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + (-1.28 - 5.23i)T \) |
| good | 2 | \( 1 + (-0.234 - 0.536i)T + (-1.36 + 1.46i)T^{2} \) |
| 5 | \( 1 + (-1.79 + 0.269i)T + (4.77 - 1.47i)T^{2} \) |
| 7 | \( 1 + (-2.59 + 2.40i)T + (0.523 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-0.974 - 0.838i)T + (1.63 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.06 + 3.02i)T + (-4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-2.82 + 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 + (5.81 - 3.65i)T + (8.24 - 17.1i)T^{2} \) |
| 23 | \( 1 + (8.33 - 3.27i)T + (16.8 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.72 - 2.74i)T + (9.13 + 29.6i)T^{2} \) |
| 37 | \( 1 + (-2.62 + 7.49i)T + (-28.9 - 23.0i)T^{2} \) |
| 41 | \( 1 + (6.95 - 1.86i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.32 - 1.79i)T + (-12.6 + 41.0i)T^{2} \) |
| 47 | \( 1 + (1.08 - 1.26i)T + (-7.00 - 46.4i)T^{2} \) |
| 53 | \( 1 + (6.55 - 5.22i)T + (11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 + (-7.47 - 4.31i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-13.5 - 0.506i)T + (60.8 + 4.55i)T^{2} \) |
| 67 | \( 1 + (-10.9 - 0.819i)T + (66.2 + 9.98i)T^{2} \) |
| 71 | \( 1 + (-3.89 - 1.87i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-0.963 - 8.55i)T + (-71.1 + 16.2i)T^{2} \) |
| 79 | \( 1 + (-0.335 - 1.77i)T + (-73.5 + 28.8i)T^{2} \) |
| 83 | \( 1 + (-0.302 - 0.980i)T + (-68.5 + 46.7i)T^{2} \) |
| 89 | \( 1 + (-2.15 - 0.242i)T + (86.7 + 19.8i)T^{2} \) |
| 97 | \( 1 + (0.817 - 1.54i)T + (-54.6 - 80.1i)T^{2} \) |
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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
−10.12525679574809509982505246793, −9.788051838498493678375383849492, −8.230643661159382781908078382463, −7.62006136157079168450901941464, −6.71251560080733190304495625064, −5.73445722009731912236963950213, −5.09837601362028438615505902626, −4.00016693982987628565207577291, −2.24951883128407025259987872770, −1.30005540456404295328312943212,
2.03115972647279126678143172410, 2.27192774003502227559949433850, 3.90518001411876091110519162229, 4.84424631574911541335792057393, 6.11885163815333948326567566379, 6.64676945544776247745013127329, 8.142478609573053273993985793468, 8.360319260858654856430361674423, 9.699237968553404717837398259309, 10.35983274690129947569425892643
