| L(s) = 1 | + (−2.77 + 0.417i)2-s + (−1.04 − 0.714i)3-s + (5.59 − 1.72i)4-s + (−0.263 + 3.51i)5-s + (3.20 + 1.54i)6-s + (−2.12 − 1.57i)7-s + (−9.72 + 4.68i)8-s + (−0.507 − 1.29i)9-s + (−0.737 − 9.83i)10-s + (−0.624 + 1.59i)11-s + (−7.09 − 2.18i)12-s + (0.623 + 0.781i)13-s + (6.54 + 3.48i)14-s + (2.78 − 3.49i)15-s + (15.3 − 10.4i)16-s + (1.71 − 1.59i)17-s + ⋯ |
| L(s) = 1 | + (−1.95 + 0.295i)2-s + (−0.605 − 0.412i)3-s + (2.79 − 0.862i)4-s + (−0.117 + 1.57i)5-s + (1.30 + 0.629i)6-s + (−0.803 − 0.595i)7-s + (−3.43 + 1.65i)8-s + (−0.169 − 0.431i)9-s + (−0.233 − 3.11i)10-s + (−0.188 + 0.480i)11-s + (−2.04 − 0.632i)12-s + (0.172 + 0.216i)13-s + (1.74 + 0.930i)14-s + (0.719 − 0.902i)15-s + (3.83 − 2.61i)16-s + (0.415 − 0.385i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.322370 - 0.0406023i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.322370 - 0.0406023i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.12 + 1.57i)T \) |
| 13 | \( 1 + (-0.623 - 0.781i)T \) |
| good | 2 | \( 1 + (2.77 - 0.417i)T + (1.91 - 0.589i)T^{2} \) |
| 3 | \( 1 + (1.04 + 0.714i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (0.263 - 3.51i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (0.624 - 1.59i)T + (-8.06 - 7.48i)T^{2} \) |
| 17 | \( 1 + (-1.71 + 1.59i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.726 + 1.25i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.44 + 5.98i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.18 - 5.18i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (0.377 + 0.653i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.07 + 0.638i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-5.55 + 2.67i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-6.52 - 3.14i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-4.00 + 0.603i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-6.04 + 1.86i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.756 + 10.0i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-4.30 - 1.32i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-3.13 - 5.43i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.304 + 1.33i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (2.07 + 0.313i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-5.18 + 8.98i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.20 + 11.5i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-1.15 - 2.95i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
| 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
−10.43381151063850489980151218532, −9.820715281855677259702623961497, −8.919762929238509692001264671327, −7.64004313631394864188365627375, −7.11026315940269409311960190995, −6.51058975805554378653981907025, −5.92822344143213790899316744523, −3.36215492911960664390435422047, −2.30568952085993001625930947194, −0.53105322335678886470696718411,
0.793585266210168220322934968079, 2.24952771400591353875006524635, 3.76901166423975164445816531755, 5.62330505442790437618432623880, 6.01286778829454656850046179324, 7.64133713748424459035379130267, 8.242641880017065165285285788358, 8.971904412952801821472718866551, 9.706389608683641976217308888276, 10.30013061052839483250146810807
