| L(s) = 1 | + (−1.72 − 0.195i)3-s + (0.517 + 0.188i)5-s + (0.780 − 4.42i)7-s + (2.92 + 0.671i)9-s + (1.48 − 0.541i)11-s + (−1.64 + 1.38i)13-s + (−0.853 − 0.425i)15-s + (1.17 + 2.03i)17-s + (−0.0655 + 0.113i)19-s + (−2.20 + 7.46i)21-s + (0.0132 + 0.0750i)23-s + (−3.59 − 3.01i)25-s + (−4.90 − 1.72i)27-s + (−4.07 − 3.41i)29-s + (−1.30 − 7.37i)31-s + ⋯ |
| L(s) = 1 | + (−0.993 − 0.112i)3-s + (0.231 + 0.0842i)5-s + (0.295 − 1.67i)7-s + (0.974 + 0.223i)9-s + (0.448 − 0.163i)11-s + (−0.457 + 0.383i)13-s + (−0.220 − 0.109i)15-s + (0.285 + 0.494i)17-s + (−0.0150 + 0.0260i)19-s + (−0.481 + 1.62i)21-s + (0.00276 + 0.0156i)23-s + (−0.719 − 0.603i)25-s + (−0.943 − 0.332i)27-s + (−0.756 − 0.634i)29-s + (−0.233 − 1.32i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.621356 - 0.777476i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.621356 - 0.777476i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.72 + 0.195i)T \) |
| good | 5 | \( 1 + (-0.517 - 0.188i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.780 + 4.42i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.48 + 0.541i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.64 - 1.38i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.17 - 2.03i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0655 - 0.113i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0132 - 0.0750i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.07 + 3.41i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.30 + 7.37i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.49 - 2.58i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.65 + 7.26i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.587 + 0.213i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.384 + 2.18i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 + (5.48 + 1.99i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.08 + 11.8i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (10.0 - 8.47i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (6.94 + 12.0i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.48 + 2.58i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.90 + 1.60i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.50 - 2.93i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (0.417 - 0.723i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12.6 - 4.60i)T + (74.3 - 62.3i)T^{2} \) |
| 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
−10.11283553510330933714286073646, −9.382262239804222930726259160821, −7.86349514336820612546701048946, −7.34639150490224707018203550576, −6.44127633686073599814093372372, −5.64957878674630723132649717750, −4.38108719746887342593262783478, −3.91525279146035792643535893701, −1.88742997312366145749464649791, −0.57259331931070212381492921763,
1.53574455533649591347352641990, 2.84499075471512006235917091102, 4.36125119523403923288807989033, 5.48400579818397139979444364806, 5.66881167372187383793556497086, 6.84986713120287017758537049800, 7.77981078708764512084826775584, 9.027307646421591373705259669214, 9.445477749486030021329964272428, 10.46619244206170896331204670055
