| L(s) = 1 | + (−0.698 − 0.123i)2-s + (20.3 + 17.7i)3-s + (−59.6 − 21.7i)4-s + (95.6 + 113. i)5-s + (−12.0 − 14.9i)6-s + (−285. + 103. i)7-s + (78.3 + 45.2i)8-s + (96.7 + 722. i)9-s + (−52.7 − 91.3i)10-s + (−448. + 534. i)11-s + (−826. − 1.50e3i)12-s + (509. + 2.88e3i)13-s + (212. − 37.4i)14-s + (−83.1 + 4.01e3i)15-s + (3.06e3 + 2.57e3i)16-s + (−3.11e3 + 1.79e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.0873 − 0.0153i)2-s + (0.752 + 0.658i)3-s + (−0.932 − 0.339i)4-s + (0.764 + 0.911i)5-s + (−0.0555 − 0.0690i)6-s + (−0.832 + 0.303i)7-s + (0.152 + 0.0883i)8-s + (0.132 + 0.991i)9-s + (−0.0527 − 0.0913i)10-s + (−0.336 + 0.401i)11-s + (−0.478 − 0.869i)12-s + (0.231 + 1.31i)13-s + (0.0773 − 0.0136i)14-s + (−0.0246 + 1.18i)15-s + (0.748 + 0.627i)16-s + (−0.633 + 0.365i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.918777 + 1.15729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.918777 + 1.15729i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-20.3 - 17.7i)T \) |
| good | 2 | \( 1 + (0.698 + 0.123i)T + (60.1 + 21.8i)T^{2} \) |
| 5 | \( 1 + (-95.6 - 113. i)T + (-2.71e3 + 1.53e4i)T^{2} \) |
| 7 | \( 1 + (285. - 103. i)T + (9.01e4 - 7.56e4i)T^{2} \) |
| 11 | \( 1 + (448. - 534. i)T + (-3.07e5 - 1.74e6i)T^{2} \) |
| 13 | \( 1 + (-509. - 2.88e3i)T + (-4.53e6 + 1.65e6i)T^{2} \) |
| 17 | \( 1 + (3.11e3 - 1.79e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-4.61e3 + 7.99e3i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-7.85e3 + 2.15e4i)T + (-1.13e8 - 9.51e7i)T^{2} \) |
| 29 | \( 1 + (-1.94e4 - 3.42e3i)T + (5.58e8 + 2.03e8i)T^{2} \) |
| 31 | \( 1 + (-4.80e3 - 1.74e3i)T + (6.79e8 + 5.70e8i)T^{2} \) |
| 37 | \( 1 + (-7.32e3 - 1.26e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (1.90e4 - 3.35e3i)T + (4.46e9 - 1.62e9i)T^{2} \) |
| 43 | \( 1 + (-1.10e5 - 9.29e4i)T + (1.09e9 + 6.22e9i)T^{2} \) |
| 47 | \( 1 + (1.39e3 + 3.83e3i)T + (-8.25e9 + 6.92e9i)T^{2} \) |
| 53 | \( 1 + 1.80e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-1.73e5 - 2.07e5i)T + (-7.32e9 + 4.15e10i)T^{2} \) |
| 61 | \( 1 + (1.29e5 - 4.69e4i)T + (3.94e10 - 3.31e10i)T^{2} \) |
| 67 | \( 1 + (-1.02e4 - 5.78e4i)T + (-8.50e10 + 3.09e10i)T^{2} \) |
| 71 | \( 1 + (2.05e5 - 1.18e5i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (-2.11e5 + 3.65e5i)T + (-7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (3.28e4 - 1.86e5i)T + (-2.28e11 - 8.31e10i)T^{2} \) |
| 83 | \( 1 + (-7.57e5 - 1.33e5i)T + (3.07e11 + 1.11e11i)T^{2} \) |
| 89 | \( 1 + (8.35e5 + 4.82e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (-1.19e6 - 1.00e6i)T + (1.44e11 + 8.20e11i)T^{2} \) |
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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
−16.22304980298092642850845391862, −14.89769115461049387050207912845, −14.02196893423998630971242165336, −13.09563773321623590058645603271, −10.70023779045738467912301462931, −9.677867596498864548615305212779, −8.828446062978237835007140268756, −6.57211111343127118763223542853, −4.57764854979561784574286590662, −2.63593367198022911587396569889,
0.857538616887083412294377880335, 3.37142174333377133299333390633, 5.58880023407254327229130450129, 7.71642963415135658488187489319, 8.946958612509028558319028488376, 9.877702747018239101118963941912, 12.49794629353305759483766016879, 13.28607467734570921387385939021, 13.84967936176226846065502526685, 15.73352296927837285235346848928
