| L(s) = 1 | + (−1.19 − 1.60i)2-s + (2.91 + 0.712i)3-s + (−1.16 + 3.82i)4-s + (−3.09 + 1.12i)5-s + (−2.32 − 5.53i)6-s + (9.81 − 1.73i)7-s + (7.53 − 2.69i)8-s + (7.98 + 4.15i)9-s + (5.49 + 3.62i)10-s + (0.511 − 1.40i)11-s + (−6.10 + 10.3i)12-s + (7.50 + 6.29i)13-s + (−14.4 − 13.6i)14-s + (−9.82 + 1.07i)15-s + (−13.3 − 8.88i)16-s + (2.21 − 3.83i)17-s + ⋯ |
| L(s) = 1 | + (−0.595 − 0.803i)2-s + (0.971 + 0.237i)3-s + (−0.290 + 0.957i)4-s + (−0.619 + 0.225i)5-s + (−0.387 − 0.921i)6-s + (1.40 − 0.247i)7-s + (0.941 − 0.337i)8-s + (0.887 + 0.461i)9-s + (0.549 + 0.362i)10-s + (0.0464 − 0.127i)11-s + (−0.509 + 0.860i)12-s + (0.577 + 0.484i)13-s + (−1.03 − 0.978i)14-s + (−0.654 + 0.0718i)15-s + (−0.831 − 0.555i)16-s + (0.130 − 0.225i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.36530 - 0.299559i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36530 - 0.299559i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.19 + 1.60i)T \) |
| 3 | \( 1 + (-2.91 - 0.712i)T \) |
| good | 5 | \( 1 + (3.09 - 1.12i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-9.81 + 1.73i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-0.511 + 1.40i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-7.50 - 6.29i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-2.21 + 3.83i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-15.6 + 9.02i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (31.4 + 5.54i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-17.2 + 14.4i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (42.5 + 7.50i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (35.6 - 61.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (2.00 + 1.68i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-15.2 + 41.8i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-8.16 + 1.43i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 54.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-31.7 - 87.3i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (14.4 + 81.7i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-45.3 + 53.9i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (95.6 + 55.2i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (28.1 + 48.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (68.8 + 82.0i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (21.4 + 25.5i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-9.14 - 15.8i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (69.4 + 25.2i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−13.58141621692163737111787150759, −11.99909652841266308102219257056, −11.25536362270866266307705986233, −10.23318761761180241672238019690, −9.004538807867704092605738244668, −8.090461133472223197013951689123, −7.38584072314295522417657128031, −4.59344103163320824974302250920, −3.49961731879532121780758403039, −1.76540739039368792951140742732,
1.60412541844166636733230124435, 4.08870968045902286254342567786, 5.59660703826625581557377121507, 7.37715432399395619054480449043, 8.077060511485563051371257686348, 8.746290687531152306843784539379, 10.04947463328418459360663179137, 11.32803182992325744458197928501, 12.61766150683720911611909740663, 14.15555024566799772807312545023
