| L(s) = 1 | + (−1.59 + 2.54i)3-s + (−2.52 + 1.87i)5-s + (−4.60 + 3.03i)7-s + (−3.93 − 8.09i)9-s + (0.644 − 5.51i)11-s + (3.12 + 10.4i)13-s + (−0.764 − 9.40i)15-s + (−9.72 − 1.71i)17-s + (−5.56 − 31.5i)19-s + (−0.379 − 16.5i)21-s + (11.1 − 16.9i)23-s + (−4.33 + 14.4i)25-s + (26.8 + 2.85i)27-s + (18.8 − 17.7i)29-s + (1.17 − 20.1i)31-s + ⋯ |
| L(s) = 1 | + (−0.530 + 0.847i)3-s + (−0.504 + 0.375i)5-s + (−0.658 + 0.432i)7-s + (−0.437 − 0.899i)9-s + (0.0586 − 0.501i)11-s + (0.240 + 0.803i)13-s + (−0.0509 − 0.626i)15-s + (−0.572 − 0.100i)17-s + (−0.292 − 1.66i)19-s + (−0.0180 − 0.787i)21-s + (0.485 − 0.738i)23-s + (−0.173 + 0.578i)25-s + (0.994 + 0.105i)27-s + (0.649 − 0.612i)29-s + (0.0378 − 0.649i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.157996 - 0.189310i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.157996 - 0.189310i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.59 - 2.54i)T \) |
| good | 5 | \( 1 + (2.52 - 1.87i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (4.60 - 3.03i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (-0.644 + 5.51i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (-3.12 - 10.4i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (9.72 + 1.71i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (5.56 + 31.5i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (-11.1 + 16.9i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (-18.8 + 17.7i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (-1.17 + 20.1i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (51.6 - 18.7i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (-14.2 + 60.0i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (8.52 + 19.7i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (52.0 - 3.03i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (8.55 + 4.94i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (9.04 + 77.3i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (-19.5 - 9.84i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (42.3 - 44.8i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (61.2 - 72.9i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (46.6 - 39.1i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-10.3 + 2.44i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (18.8 + 79.3i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (-15.4 - 18.4i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (91.4 - 122. i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−11.24922034015488120545887384243, −10.32490108622356195717002122679, −9.203715200920925871753194819168, −8.677520826735629938321019074019, −6.96348908861914216618529934094, −6.29996212539666327470177948603, −5.01583035023297748923487299591, −3.95305273409959343927174250254, −2.80702444287382931873507704314, −0.12716726392059312264689252830,
1.46445313582843789586466398939, 3.29493182749962378571940844812, 4.69024054532194608893594207317, 5.91258908014358230481450766670, 6.83067205697509633113394998848, 7.78580829438885105553681398083, 8.571364722423323200826170102592, 10.01400986842519823912818330649, 10.75094326006504366540634161972, 11.88089702560392196129000970113
