| L(s) = 1 | + (−1.69 − 0.336i)3-s + (−0.202 + 1.41i)5-s + (−0.147 − 1.54i)7-s + (2.77 + 1.14i)9-s + (−2.41 + 0.968i)11-s + (−1.91 − 2.01i)13-s + (0.819 − 2.32i)15-s + (3.53 − 1.22i)17-s + (−0.0830 − 0.00792i)19-s + (−0.269 + 2.67i)21-s + (0.357 − 0.0170i)23-s + (2.84 + 0.836i)25-s + (−4.32 − 2.87i)27-s + (2.14 − 1.23i)29-s + (2.40 − 2.51i)31-s + ⋯ |
| L(s) = 1 | + (−0.980 − 0.194i)3-s + (−0.0907 + 0.630i)5-s + (−0.0556 − 0.582i)7-s + (0.924 + 0.381i)9-s + (−0.729 + 0.292i)11-s + (−0.532 − 0.558i)13-s + (0.211 − 0.601i)15-s + (0.858 − 0.297i)17-s + (−0.0190 − 0.00181i)19-s + (−0.0587 + 0.582i)21-s + (0.0744 − 0.00354i)23-s + (0.569 + 0.167i)25-s + (−0.832 − 0.553i)27-s + (0.398 − 0.230i)29-s + (0.431 − 0.452i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.803890 - 0.425061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.803890 - 0.425061i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.69 + 0.336i)T \) |
| 67 | \( 1 + (-8.16 - 0.605i)T \) |
| good | 5 | \( 1 + (0.202 - 1.41i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (0.147 + 1.54i)T + (-6.87 + 1.32i)T^{2} \) |
| 11 | \( 1 + (2.41 - 0.968i)T + (7.96 - 7.59i)T^{2} \) |
| 13 | \( 1 + (1.91 + 2.01i)T + (-0.618 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-3.53 + 1.22i)T + (13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (0.0830 + 0.00792i)T + (18.6 + 3.59i)T^{2} \) |
| 23 | \( 1 + (-0.357 + 0.0170i)T + (22.8 - 2.18i)T^{2} \) |
| 29 | \( 1 + (-2.14 + 1.23i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.40 + 2.51i)T + (-1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-3.28 + 5.68i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.45 - 0.280i)T + (38.0 + 15.2i)T^{2} \) |
| 43 | \( 1 + (6.93 + 6.00i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-5.86 + 11.3i)T + (-27.2 - 38.2i)T^{2} \) |
| 53 | \( 1 + (-1.34 - 1.55i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (0.862 + 2.93i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-4.16 + 10.3i)T + (-44.1 - 42.0i)T^{2} \) |
| 71 | \( 1 + (-3.54 - 1.22i)T + (55.8 + 43.8i)T^{2} \) |
| 73 | \( 1 + (3.48 + 1.39i)T + (52.8 + 50.3i)T^{2} \) |
| 79 | \( 1 + (2.37 - 0.577i)T + (70.2 - 36.1i)T^{2} \) |
| 83 | \( 1 + (-3.35 + 4.26i)T + (-19.5 - 80.6i)T^{2} \) |
| 89 | \( 1 + (4.10 - 6.39i)T + (-36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-1.05 - 0.609i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.33285384801455277362779401707, −9.666156059350792030321203915185, −8.131457398879976409752251915723, −7.34063640645591708536748848512, −6.80248302008779189910651352235, −5.63939329955460476753614689988, −4.94083667806467137072284026287, −3.72151980798576708651967729825, −2.39980601669304470260245695973, −0.61398984716898203286506058126,
1.14811897929698902414880280915, 2.84003935954926817388632535665, 4.33567481895215094351744219995, 5.09271793228935436003713632957, 5.84131412624637434413788542488, 6.78022178028490398244128133778, 7.85139283648689703037788986624, 8.747799793288693595372313387825, 9.682752680064034005209933135327, 10.37496050533643578796401821864
