| L(s) = 1 | + (0.753 − 0.274i)2-s + (−1.68 + 0.386i)3-s + (−1.03 + 0.872i)4-s + (−0.477 − 2.70i)5-s + (−1.16 + 0.753i)6-s + (1.82 + 1.52i)7-s + (−1.34 + 2.33i)8-s + (2.70 − 1.30i)9-s + (−1.10 − 1.90i)10-s + (−0.0434 + 0.246i)11-s + (1.41 − 1.87i)12-s + (−2.45 − 0.893i)13-s + (1.79 + 0.651i)14-s + (1.85 + 4.38i)15-s + (0.0969 − 0.549i)16-s + (0.146 + 0.254i)17-s + ⋯ |
| L(s) = 1 | + (0.532 − 0.193i)2-s + (−0.974 + 0.223i)3-s + (−0.519 + 0.436i)4-s + (−0.213 − 1.21i)5-s + (−0.475 + 0.307i)6-s + (0.688 + 0.577i)7-s + (−0.475 + 0.823i)8-s + (0.900 − 0.434i)9-s + (−0.348 − 0.603i)10-s + (−0.0130 + 0.0742i)11-s + (0.409 − 0.541i)12-s + (−0.680 − 0.247i)13-s + (0.478 + 0.174i)14-s + (0.478 + 1.13i)15-s + (0.0242 − 0.137i)16-s + (0.0355 + 0.0616i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.628905 - 0.0256406i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.628905 - 0.0256406i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.68 - 0.386i)T \) |
| good | 2 | \( 1 + (-0.753 + 0.274i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (0.477 + 2.70i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.82 - 1.52i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.0434 - 0.246i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (2.45 + 0.893i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.146 - 0.254i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.39 + 2.41i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.12 - 4.30i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.333 + 0.121i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.11 + 1.77i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.49 - 6.05i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.13 - 3.32i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.0452 + 0.256i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (8.75 + 7.34i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 5.43T + 53T^{2} \) |
| 59 | \( 1 + (-1.03 - 5.88i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (9.07 + 7.61i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (1.70 + 0.619i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.185 - 0.320i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.51 - 4.35i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.754 - 0.274i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.58 + 0.942i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (5.22 - 9.05i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.57 - 14.6i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−17.39310865614492276094368590419, −16.36609123493414288130969298140, −15.04056456573727062257644577498, −13.32255920859715428834748237753, −12.24561701546880513039357396736, −11.60070668613942952650833149183, −9.510351887852142116256011477018, −8.066773899046704640873529559702, −5.43053834861944240826228762655, −4.50233024352238829707502863827,
4.42756300253571929297504477448, 6.09892538076565959197579467777, 7.44067986238791737220535405692, 10.04609526259177094802274295503, 11.03374024552937995351347276148, 12.41929454777779246360061655693, 14.00428965540572138329800514180, 14.70523764281855525226493452263, 16.16732841791044658656689177043, 17.68927672098519023405308213415
