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
−11.88089702560392196129000970113, −10.75094326006504366540634161972, −10.01400986842519823912818330649, −8.571364722423323200826170102592, −7.78580829438885105553681398083, −6.83067205697509633113394998848, −5.91258908014358230481450766670, −4.69024054532194608893594207317, −3.29493182749962378571940844812, −1.46445313582843789586466398939,
0.12716726392059312264689252830, 2.80702444287382931873507704314, 3.95305273409959343927174250254, 5.01583035023297748923487299591, 6.29996212539666327470177948603, 6.96348908861914216618529934094, 8.677520826735629938321019074019, 9.203715200920925871753194819168, 10.32490108622356195717002122679, 11.24922034015488120545887384243