| L(s) = 1 | + (−0.524 − 1.44i)2-s + (−2.15 − 2.08i)3-s + (1.26 − 1.05i)4-s + (−0.0496 + 0.00875i)5-s + (−1.86 + 4.20i)6-s + (7.55 + 6.33i)7-s + (−7.50 − 4.33i)8-s + (0.320 + 8.99i)9-s + (0.0386 + 0.0669i)10-s + (12.0 + 2.12i)11-s + (−4.93 − 0.343i)12-s + (−11.4 − 4.18i)13-s + (5.17 − 14.2i)14-s + (0.125 + 0.0845i)15-s + (−1.16 + 6.58i)16-s + (−12.7 + 7.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.262 − 0.720i)2-s + (−0.719 − 0.694i)3-s + (0.315 − 0.264i)4-s + (−0.00992 + 0.00175i)5-s + (−0.311 + 0.700i)6-s + (1.07 + 0.905i)7-s + (−0.937 − 0.541i)8-s + (0.0355 + 0.999i)9-s + (0.00386 + 0.00669i)10-s + (1.09 + 0.193i)11-s + (−0.411 − 0.0286i)12-s + (−0.883 − 0.321i)13-s + (0.369 − 1.01i)14-s + (0.00836 + 0.00563i)15-s + (−0.0725 + 0.411i)16-s + (−0.752 + 0.434i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0823 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0823 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.589457 - 0.542744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.589457 - 0.542744i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.15 + 2.08i)T \) |
| good | 2 | \( 1 + (0.524 + 1.44i)T + (-3.06 + 2.57i)T^{2} \) |
| 5 | \( 1 + (0.0496 - 0.00875i)T + (23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-7.55 - 6.33i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-12.0 - 2.12i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (11.4 + 4.18i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (12.7 - 7.38i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-6.26 + 10.8i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-5.93 - 7.07i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-15.1 - 41.6i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (24.2 - 20.3i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (7.19 + 12.4i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-23.5 + 64.8i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-7.01 + 39.8i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-1.24 + 1.48i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 47.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (72.6 - 12.8i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (4.54 + 3.81i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-45.8 - 16.6i)T + (3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (8.61 - 4.97i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (44.2 - 76.6i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-105. + 38.3i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (6.65 + 18.2i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (33.9 + 19.5i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (0.0670 - 0.380i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−17.37270415022112720154908430453, −15.59462984013711068642995336065, −14.40571582391525044093116871531, −12.47259286526648206002642639282, −11.72380841754890144354779853632, −10.78002170596059864272456748783, −8.999745200807834041372181398094, −7.03984874061305211481221686035, −5.40683420046705169116234452266, −1.88608959918198157315690167946,
4.40980377017705873201313002849, 6.33590015079621082912787368235, 7.75289444874784656875334542904, 9.480518208941857757363406871145, 11.18014018667153589249265914498, 11.92784264554128104825436683688, 14.20374970738409738138644013030, 15.18241680993172674981198566400, 16.51611854671488929306478381857, 17.14203096055655788136712989823
