| L(s) = 1 | + (−0.463 − 2.96i)3-s + (−2.73 + 2.03i)5-s + (−1.12 + 0.737i)7-s + (−8.57 + 2.74i)9-s + (−0.325 + 2.78i)11-s + (6.15 + 20.5i)13-s + (7.29 + 7.15i)15-s + (5.48 + 0.967i)17-s + (0.824 + 4.67i)19-s + (2.70 + 2.98i)21-s + (10.9 − 16.6i)23-s + (−3.84 + 12.8i)25-s + (12.1 + 24.1i)27-s + (−14.1 + 13.3i)29-s + (−0.486 + 8.35i)31-s + ⋯ |
| L(s) = 1 | + (−0.154 − 0.987i)3-s + (−0.546 + 0.406i)5-s + (−0.160 + 0.105i)7-s + (−0.952 + 0.305i)9-s + (−0.0295 + 0.253i)11-s + (0.473 + 1.58i)13-s + (0.486 + 0.476i)15-s + (0.322 + 0.0569i)17-s + (0.0434 + 0.246i)19-s + (0.128 + 0.142i)21-s + (0.475 − 0.722i)23-s + (−0.153 + 0.513i)25-s + (0.448 + 0.893i)27-s + (−0.488 + 0.460i)29-s + (−0.0157 + 0.269i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.830177 + 0.512104i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.830177 + 0.512104i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.463 + 2.96i)T \) |
| good | 5 | \( 1 + (2.73 - 2.03i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (1.12 - 0.737i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (0.325 - 2.78i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (-6.15 - 20.5i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (-5.48 - 0.967i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (-0.824 - 4.67i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (-10.9 + 16.6i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (14.1 - 13.3i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (0.486 - 8.35i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (-34.4 + 12.5i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (6.96 - 29.3i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (-24.2 - 56.1i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-16.7 + 0.974i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (84.4 + 48.7i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-4.20 - 35.9i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (-0.599 - 0.301i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (-23.3 + 24.7i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (56.0 - 66.8i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (70.3 - 59.0i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (31.3 - 7.43i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (12.1 + 51.0i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (75.7 + 90.2i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (34.4 - 46.2i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−11.45940708549477697577833448238, −11.07916458131745732568179315543, −9.586909976646937723051393862216, −8.604238422750257508234844932197, −7.56964764853530904931236081570, −6.81321067522199663752681296571, −5.93922543073449944699728379353, −4.43006257386518891393206730509, −3.00890282513474372706223271192, −1.53017091865621372183395886371,
0.48997824125248561224375188306, 3.05214155193005099919443361004, 4.00285819705863770051618540730, 5.19507347533180863151097996790, 6.02252274216736741018428862532, 7.64393415731145340843652768259, 8.455220735336213605740138457023, 9.431841433406656503920357034420, 10.36583035564951162279043675305, 11.10745144839024281693842863249
