| L(s) = 1 | + (−1.68 + 0.297i)2-s + (−1.11 − 1.32i)3-s + (−0.997 + 0.362i)4-s + (8.34 + 3.03i)5-s + (2.27 + 1.90i)6-s + (5.55 − 9.61i)7-s + (7.51 − 4.33i)8-s + (−0.520 + 2.95i)9-s + (−14.9 − 2.64i)10-s + (4.39 + 7.61i)11-s + (1.59 + 0.918i)12-s + (−3.70 + 4.41i)13-s + (−6.51 + 17.8i)14-s + (−5.26 − 14.4i)15-s + (−8.14 + 6.83i)16-s + (0.256 + 1.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.844 + 0.148i)2-s + (−0.371 − 0.442i)3-s + (−0.249 + 0.0907i)4-s + (1.66 + 0.607i)5-s + (0.379 + 0.318i)6-s + (0.793 − 1.37i)7-s + (0.939 − 0.542i)8-s + (−0.0578 + 0.328i)9-s + (−1.49 − 0.264i)10-s + (0.399 + 0.692i)11-s + (0.132 + 0.0765i)12-s + (−0.284 + 0.339i)13-s + (−0.465 + 1.27i)14-s + (−0.350 − 0.963i)15-s + (−0.508 + 0.427i)16-s + (0.0150 + 0.0855i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.851852 - 0.0628337i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.851852 - 0.0628337i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.11 + 1.32i)T \) |
| 19 | \( 1 + (6.57 + 17.8i)T \) |
| good | 2 | \( 1 + (1.68 - 0.297i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-8.34 - 3.03i)T + (19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (-5.55 + 9.61i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-4.39 - 7.61i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (3.70 - 4.41i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (-0.256 - 1.45i)T + (-271. + 98.8i)T^{2} \) |
| 23 | \( 1 + (-9.77 + 3.55i)T + (405. - 340. i)T^{2} \) |
| 29 | \( 1 + (43.0 + 7.59i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (-7.87 - 4.54i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 14.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-10.3 - 12.3i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (56.1 + 20.4i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (12.9 - 73.3i)T + (-2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + (1.71 + 4.71i)T + (-2.15e3 + 1.80e3i)T^{2} \) |
| 59 | \( 1 + (36.2 - 6.39i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (64.3 - 23.4i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-39.1 - 6.89i)T + (4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-2.05 + 5.64i)T + (-3.86e3 - 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-47.9 + 40.2i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (86.9 + 103. i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (22.0 - 38.1i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (16.7 - 19.9i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-45.8 + 8.08i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−14.65976221828178928954646781344, −13.73780014303069134061443836050, −13.06696335476969189737911617594, −11.06116607047426709465680868579, −10.19064107003326996885326180024, −9.236347506071681231612056282050, −7.51717571059986993335568135718, −6.64480026998855245772458539055, −4.72502168660705151135644087606, −1.58907048704164355690021250967,
1.78603123453533487004137727402, 5.12649984736952009257011006717, 5.81818715263862077616124529085, 8.438977957873685433775825182656, 9.180870805408188903989376476071, 10.03580670931772903887878458526, 11.28370425899231055233536249501, 12.71571663280609913206340815978, 13.96469065917934432819234392562, 14.93609840167497661768053082865
