| L(s) = 1 | + (−1.28 + 3.53i)2-s + (2.97 + 0.416i)3-s + (−7.76 − 6.51i)4-s + (−0.526 − 0.0928i)5-s + (−5.29 + 9.95i)6-s + (2.13 − 1.79i)7-s + (19.9 − 11.5i)8-s + (8.65 + 2.47i)9-s + (1.00 − 1.74i)10-s + (−7.23 + 1.27i)11-s + (−20.3 − 22.5i)12-s + (5.53 − 2.01i)13-s + (3.58 + 9.83i)14-s + (−1.52 − 0.495i)15-s + (8.00 + 45.3i)16-s + (−20.4 − 11.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.642 + 1.76i)2-s + (0.990 + 0.138i)3-s + (−1.94 − 1.62i)4-s + (−0.105 − 0.0185i)5-s + (−0.882 + 1.65i)6-s + (0.304 − 0.255i)7-s + (2.49 − 1.44i)8-s + (0.961 + 0.275i)9-s + (0.100 − 0.174i)10-s + (−0.657 + 0.115i)11-s + (−1.69 − 1.88i)12-s + (0.425 − 0.154i)13-s + (0.255 + 0.702i)14-s + (−0.101 − 0.0330i)15-s + (0.500 + 2.83i)16-s + (−1.20 − 0.696i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.488190 + 0.689086i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.488190 + 0.689086i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.97 - 0.416i)T \) |
| good | 2 | \( 1 + (1.28 - 3.53i)T + (-3.06 - 2.57i)T^{2} \) |
| 5 | \( 1 + (0.526 + 0.0928i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-2.13 + 1.79i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (7.23 - 1.27i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-5.53 + 2.01i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (20.4 + 11.8i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (3.60 + 6.23i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-4.73 + 5.63i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (6.46 - 17.7i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-32.2 - 27.0i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (27.6 - 47.8i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (10.8 + 29.7i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (1.28 + 7.27i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-37.7 - 44.9i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 69.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-30.9 - 5.44i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (27.3 - 22.9i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-75.2 + 27.3i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-13.3 - 7.70i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (47.6 + 82.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (1.95 + 0.710i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (11.1 - 30.6i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-69.3 + 40.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (2.66 + 15.1i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−17.43725756445957688765929300212, −15.89045264969932987974362448996, −15.45567207351732537375417259036, −14.15940894440607416329076229462, −13.34789744538924778080933458071, −10.37882953682330559320971244840, −9.003299911799428736172928279830, −8.050577212320809407032717148668, −6.84570153881930367892686819101, −4.75428502584780368783668853422,
2.19051274152377244648913243771, 3.97321879849869310987967145559, 7.995113482372864151832998842337, 8.964830530242115235850347045254, 10.24462626065951346400047172551, 11.50167267973161578195210577755, 12.88691003100176595512424559316, 13.69746621706463573212421630483, 15.42981438029858413403425008580, 17.42190141145611885383255460172
