| L(s) = 1 | + (−0.826 + 0.563i)2-s + (1.35 − 1.07i)3-s + (0.365 − 0.930i)4-s + (0.455 + 1.99i)5-s + (−0.517 + 1.65i)6-s + (0.532 + 2.59i)7-s + (0.222 + 0.974i)8-s + (0.691 − 2.91i)9-s + (−1.49 − 1.39i)10-s + (−3.34 − 1.60i)11-s + (−0.503 − 1.65i)12-s + (0.268 − 0.182i)13-s + (−1.89 − 1.84i)14-s + (2.76 + 2.22i)15-s + (−0.733 − 0.680i)16-s + (1.49 + 3.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.584 + 0.398i)2-s + (0.784 − 0.620i)3-s + (0.182 − 0.465i)4-s + (0.203 + 0.891i)5-s + (−0.211 + 0.674i)6-s + (0.201 + 0.979i)7-s + (0.0786 + 0.344i)8-s + (0.230 − 0.973i)9-s + (−0.474 − 0.439i)10-s + (−1.00 − 0.485i)11-s + (−0.145 − 0.478i)12-s + (0.0744 − 0.0507i)13-s + (−0.507 − 0.492i)14-s + (0.712 + 0.573i)15-s + (−0.183 − 0.170i)16-s + (0.362 + 0.923i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28171 + 0.838766i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28171 + 0.838766i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.826 - 0.563i)T \) |
| 3 | \( 1 + (-1.35 + 1.07i)T \) |
| 7 | \( 1 + (-0.532 - 2.59i)T \) |
| good | 5 | \( 1 + (-0.455 - 1.99i)T + (-4.50 + 2.16i)T^{2} \) |
| 11 | \( 1 + (3.34 + 1.60i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (-0.268 + 0.182i)T + (4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-1.49 - 3.80i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-3.44 - 5.97i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.45 - 6.84i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (5.70 + 0.859i)T + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 + (1.16 + 2.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.39 - 0.512i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (6.49 + 2.00i)T + (33.8 + 23.0i)T^{2} \) |
| 43 | \( 1 + (-9.00 + 2.77i)T + (35.5 - 24.2i)T^{2} \) |
| 47 | \( 1 + (2.21 - 1.51i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-3.73 + 0.562i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (4.62 - 1.42i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (-5.66 - 14.4i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (3.05 + 5.29i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.32 - 4.16i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (0.699 - 9.33i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-7.13 + 12.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.57 - 1.75i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (0.536 + 0.366i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (5.26 + 9.11i)T + (-48.5 + 84.0i)T^{2} \) |
| 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
−10.09580467426361497839047351389, −9.316536808224726413960082295009, −8.479045345795987635311244103235, −7.76395014009744625372297493981, −7.18429582748130717714850809011, −5.94641518914736152133290397386, −5.56427815525115965884519217320, −3.53214630354964065619683333443, −2.67164053239142942405505482741, −1.57020643581608674149869036521,
0.862634964600355436709814361537, 2.38531708124315729952982575138, 3.37507860687621263582367005861, 4.77904166906738544474022819455, 4.98354331486826219384090278589, 7.02367603253590094778365509483, 7.61663396367436166061660228856, 8.497823695480674561177586300669, 9.287542706531370068446543772196, 9.760998834212390966188283855255
