| L(s) = 1 | + (1.20 − 1.01i)2-s + (−10.0 − 11.9i)3-s + (−5.12 + 29.0i)4-s + (−55.5 + 20.2i)5-s + (−24.2 − 4.22i)6-s + (11.4 + 64.7i)7-s + (48.4 + 83.9i)8-s + (−41.4 + 239. i)9-s + (−46.6 + 80.7i)10-s + (−501. − 182. i)11-s + (398. − 230. i)12-s + (−361. − 303. i)13-s + (79.3 + 66.5i)14-s + (799. + 459. i)15-s + (−743. − 270. i)16-s + (434. − 752. i)17-s + ⋯ |
| L(s) = 1 | + (0.213 − 0.179i)2-s + (−0.644 − 0.764i)3-s + (−0.160 + 0.908i)4-s + (−0.994 + 0.361i)5-s + (−0.274 − 0.0479i)6-s + (0.0880 + 0.499i)7-s + (0.267 + 0.463i)8-s + (−0.170 + 0.985i)9-s + (−0.147 + 0.255i)10-s + (−1.24 − 0.454i)11-s + (0.798 − 0.462i)12-s + (−0.592 − 0.497i)13-s + (0.108 + 0.0907i)14-s + (0.916 + 0.527i)15-s + (−0.726 − 0.264i)16-s + (0.364 − 0.631i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.159284 + 0.371759i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.159284 + 0.371759i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (10.0 + 11.9i)T \) |
| good | 2 | \( 1 + (-1.20 + 1.01i)T + (5.55 - 31.5i)T^{2} \) |
| 5 | \( 1 + (55.5 - 20.2i)T + (2.39e3 - 2.00e3i)T^{2} \) |
| 7 | \( 1 + (-11.4 - 64.7i)T + (-1.57e4 + 5.74e3i)T^{2} \) |
| 11 | \( 1 + (501. + 182. i)T + (1.23e5 + 1.03e5i)T^{2} \) |
| 13 | \( 1 + (361. + 303. i)T + (6.44e4 + 3.65e5i)T^{2} \) |
| 17 | \( 1 + (-434. + 752. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.29e3 - 2.23e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-414. + 2.34e3i)T + (-6.04e6 - 2.20e6i)T^{2} \) |
| 29 | \( 1 + (1.44e3 - 1.21e3i)T + (3.56e6 - 2.01e7i)T^{2} \) |
| 31 | \( 1 + (1.41e3 - 8.01e3i)T + (-2.69e7 - 9.79e6i)T^{2} \) |
| 37 | \( 1 + (2.00e3 - 3.46e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (2.07e3 + 1.73e3i)T + (2.01e7 + 1.14e8i)T^{2} \) |
| 43 | \( 1 + (3.18e3 + 1.15e3i)T + (1.12e8 + 9.44e7i)T^{2} \) |
| 47 | \( 1 + (-3.73e3 - 2.11e4i)T + (-2.15e8 + 7.84e7i)T^{2} \) |
| 53 | \( 1 - 1.74e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (4.48e4 - 1.63e4i)T + (5.47e8 - 4.59e8i)T^{2} \) |
| 61 | \( 1 + (7.10e3 + 4.03e4i)T + (-7.93e8 + 2.88e8i)T^{2} \) |
| 67 | \( 1 + (2.69e4 + 2.26e4i)T + (2.34e8 + 1.32e9i)T^{2} \) |
| 71 | \( 1 + (-3.51e3 + 6.08e3i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-4.34e4 - 7.51e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-2.16e4 + 1.81e4i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (5.59e4 - 4.69e4i)T + (6.84e8 - 3.87e9i)T^{2} \) |
| 89 | \( 1 + (2.80e4 + 4.85e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-4.39e4 - 1.59e4i)T + (6.57e9 + 5.51e9i)T^{2} \) |
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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
−16.72513828283419182278162604265, −15.77527917093271631900974495470, −14.01468088642240134806982922466, −12.57325680306395662239642452109, −11.99621442942394043094688471799, −10.75096452993137045992085049407, −8.164414052024409234681371329247, −7.40636517219329095021865025032, −5.23230847250714620499853416564, −2.98630073897474994275305047982,
0.26226630395514823961575942368, 4.27169049904504519192493537648, 5.37988714490374057371873561838, 7.36642259542594728847533198377, 9.461671088953082503394699771550, 10.62566358467455236430108113931, 11.77737277200140764522142299171, 13.40608411482251448459966712325, 15.10517805487746342546210248918, 15.59045894242180855706402003620
