| L(s) = 1 | + (4.64 + 1.69i)2-s + (−11.1 − 10.8i)3-s + (−5.78 − 4.85i)4-s + (9.66 − 54.8i)5-s + (−33.4 − 69.4i)6-s + (60.6 − 50.9i)7-s + (−97.7 − 169. i)8-s + (5.93 + 242. i)9-s + (137. − 238. i)10-s + (30.3 + 172. i)11-s + (11.6 + 117. i)12-s + (162. − 59.0i)13-s + (368. − 133. i)14-s + (−704. + 506. i)15-s + (−125. − 714. i)16-s + (263. − 456. i)17-s + ⋯ |
| L(s) = 1 | + (0.821 + 0.298i)2-s + (−0.715 − 0.698i)3-s + (−0.180 − 0.151i)4-s + (0.172 − 0.980i)5-s + (−0.379 − 0.787i)6-s + (0.468 − 0.392i)7-s + (−0.540 − 0.935i)8-s + (0.0244 + 0.999i)9-s + (0.435 − 0.753i)10-s + (0.0756 + 0.429i)11-s + (0.0234 + 0.234i)12-s + (0.266 − 0.0969i)13-s + (0.501 − 0.182i)14-s + (−0.808 + 0.581i)15-s + (−0.123 − 0.697i)16-s + (0.221 − 0.382i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00757 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.00757 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.12764 - 1.11912i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12764 - 1.11912i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (11.1 + 10.8i)T \) |
| good | 2 | \( 1 + (-4.64 - 1.69i)T + (24.5 + 20.5i)T^{2} \) |
| 5 | \( 1 + (-9.66 + 54.8i)T + (-2.93e3 - 1.06e3i)T^{2} \) |
| 7 | \( 1 + (-60.6 + 50.9i)T + (2.91e3 - 1.65e4i)T^{2} \) |
| 11 | \( 1 + (-30.3 - 172. i)T + (-1.51e5 + 5.50e4i)T^{2} \) |
| 13 | \( 1 + (-162. + 59.0i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-263. + 456. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (343. + 594. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-3.50e3 - 2.93e3i)T + (1.11e6 + 6.33e6i)T^{2} \) |
| 29 | \( 1 + (-4.14e3 - 1.50e3i)T + (1.57e7 + 1.31e7i)T^{2} \) |
| 31 | \( 1 + (8.17e3 + 6.86e3i)T + (4.97e6 + 2.81e7i)T^{2} \) |
| 37 | \( 1 + (-7.71e3 + 1.33e4i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-5.16e3 + 1.88e3i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 + (-1.24e3 - 7.04e3i)T + (-1.38e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (1.34e3 - 1.12e3i)T + (3.98e7 - 2.25e8i)T^{2} \) |
| 53 | \( 1 + 2.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (6.41e3 - 3.63e4i)T + (-6.71e8 - 2.44e8i)T^{2} \) |
| 61 | \( 1 + (-3.35e4 + 2.81e4i)T + (1.46e8 - 8.31e8i)T^{2} \) |
| 67 | \( 1 + (3.94e4 - 1.43e4i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (1.25e4 - 2.17e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-2.05e4 - 3.55e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (6.66e4 + 2.42e4i)T + (2.35e9 + 1.97e9i)T^{2} \) |
| 83 | \( 1 + (-5.91e4 - 2.15e4i)T + (3.01e9 + 2.53e9i)T^{2} \) |
| 89 | \( 1 + (-3.56e4 - 6.18e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (6.07e3 + 3.44e4i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
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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
−16.02795055549751635761672738533, −14.51168114552708144467544477813, −13.24042462497825927467239606384, −12.66799244269531626959431338092, −11.15238765922894141191141718582, −9.266216509217707048911380401568, −7.33926893851608372303140738203, −5.67067342015995481087718873965, −4.62142844803113117123038695365, −0.988736741771169497020197923446,
3.21089136304526335256418853071, 4.86616893851013271575224991836, 6.32133536096023203453077725103, 8.718178145822411754933571984147, 10.55179919620939548408087872045, 11.47613309942166417783477566774, 12.70252367052906844244266417002, 14.33813245310154107200632294750, 14.98400900916819975512006055602, 16.59889686845972910052539924381
