| 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
−10.35983274690129947569425892643, −9.699237968553404717837398259309, −8.360319260858654856430361674423, −8.142478609573053273993985793468, −6.64676945544776247745013127329, −6.11885163815333948326567566379, −4.84424631574911541335792057393, −3.90518001411876091110519162229, −2.27192774003502227559949433850, −2.03115972647279126678143172410,
1.30005540456404295328312943212, 2.24951883128407025259987872770, 4.00016693982987628565207577291, 5.09837601362028438615505902626, 5.73445722009731912236963950213, 6.71251560080733190304495625064, 7.62006136157079168450901941464, 8.230643661159382781908078382463, 9.788051838498493678375383849492, 10.12525679574809509982505246793
