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
−10.30013061052839483250146810807, −9.706389608683641976217308888276, −8.971904412952801821472718866551, −8.242641880017065165285285788358, −7.64133713748424459035379130267, −6.01286778829454656850046179324, −5.62330505442790437618432623880, −3.76901166423975164445816531755, −2.24952771400591353875006524635, −0.793585266210168220322934968079,
0.53105322335678886470696718411, 2.30568952085993001625930947194, 3.36215492911960664390435422047, 5.92822344143213790899316744523, 6.51058975805554378653981907025, 7.11026315940269409311960190995, 7.64004313631394864188365627375, 8.919762929238509692001264671327, 9.820715281855677259702623961497, 10.43381151063850489980151218532