L(s) = 1 | + (2.05 − 1.18i)2-s + (−4.97 + 1.5i)3-s + (−1.18 + 2.05i)4-s + (−8.44 + 8.98i)6-s + (6.38 − 3.68i)7-s + 24.6i·8-s + (22.5 − 14.9i)9-s + (2.06 + 3.58i)11-s + (2.81 − 11.9i)12-s + (−68.3 − 39.4i)13-s + (8.74 − 15.1i)14-s + (19.6 + 34.1i)16-s − 33.3i·17-s + (28.5 − 57.3i)18-s − 89.3·19-s + ⋯ |
L(s) = 1 | + (0.726 − 0.419i)2-s + (−0.957 + 0.288i)3-s + (−0.148 + 0.256i)4-s + (−0.574 + 0.611i)6-s + (0.344 − 0.199i)7-s + 1.08i·8-s + (0.833 − 0.552i)9-s + (0.0567 + 0.0982i)11-s + (0.0678 − 0.288i)12-s + (−1.45 − 0.842i)13-s + (0.166 − 0.289i)14-s + (0.307 + 0.533i)16-s − 0.475i·17-s + (0.373 − 0.750i)18-s − 1.07·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0350932 - 0.234922i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0350932 - 0.234922i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.97 - 1.5i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-2.05 + 1.18i)T + (4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (-6.38 + 3.68i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-2.06 - 3.58i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (68.3 + 39.4i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 33.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 89.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + (172. + 99.7i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-25.2 - 43.7i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-3 + 5.19i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 290. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (-26.6 + 46.1i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-258. + 149. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (362. - 209. i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 399. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-49.1 + 85.0i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-341. - 592. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (195. + 112. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 512.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 994. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (100. + 174. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (959. - 553. i)T + (2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 372.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (120. - 69.6i)T + (4.56e5 - 7.90e5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.51798661857571405043099431325, −10.64122381584368440869744228798, −9.746639643386980213857909286644, −8.297296014333885858265113167123, −7.21169537150014652247388304901, −5.79246697034304836259284980008, −4.81940249351946668265277639299, −4.04036953688974081777502482038, −2.38729560536778823213696214316, −0.083468554658726522300315065908,
1.81692273881956616180767485236, 4.18844254220483479293381316940, 4.97479886292797345066595949024, 6.02700524145191875160344736539, 6.79162021203449952306102311711, 7.959589918452634815317525623170, 9.576300650747799816321712598392, 10.30842021647982782145817935460, 11.56120369530880409585832563278, 12.26848129670267832147822431452