L(s) = 1 | + (−1.17 + 2.02i)2-s + (0.573 + 0.992i)3-s + (−1.74 − 3.02i)4-s + (0.671 − 1.16i)5-s − 2.68·6-s + 3.48·8-s + (0.842 − 1.45i)9-s + (1.57 + 2.72i)10-s + (−0.573 − 0.992i)11-s + (2 − 3.46i)12-s + 13-s + 1.53·15-s + (−0.598 + 1.03i)16-s + (−2.91 − 5.05i)17-s + (1.97 + 3.42i)18-s + (1.67 − 2.89i)19-s + ⋯ |
L(s) = 1 | + (−0.828 + 1.43i)2-s + (0.330 + 0.573i)3-s + (−0.872 − 1.51i)4-s + (0.300 − 0.520i)5-s − 1.09·6-s + 1.23·8-s + (0.280 − 0.486i)9-s + (0.497 + 0.861i)10-s + (−0.172 − 0.299i)11-s + (0.577 − 0.999i)12-s + 0.277·13-s + 0.397·15-s + (−0.149 + 0.259i)16-s + (−0.707 − 1.22i)17-s + (0.465 + 0.806i)18-s + (0.383 − 0.664i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.941538 + 0.466746i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.941538 + 0.466746i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (1.17 - 2.02i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.573 - 0.992i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.671 + 1.16i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.573 + 0.992i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2.91 + 5.05i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.67 + 2.89i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.58 + 2.74i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + (0.817 + 1.41i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.25 - 7.37i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 0.292T + 41T^{2} \) |
| 43 | \( 1 + 8.15T + 43T^{2} \) |
| 47 | \( 1 + (-5.30 + 9.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.391 - 0.677i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.32 + 10.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.05 - 5.28i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.53T + 71T^{2} \) |
| 73 | \( 1 + (-7.65 - 13.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.441 - 0.764i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + (2.86 - 4.96i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5.34T + 97T^{2} \) |
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
−10.12175947129014270040579294351, −9.588103372721924940879297306073, −8.720092581491891834962973391074, −8.436893006677318981978114781318, −7.00114149927471537423227586042, −6.60783271472176063174175302832, −5.27376040689103822706882851662, −4.66502741272592187581587477565, −3.04271063713227250216556200830, −0.821294400927533823881995176095,
1.40046142178110745962061108358, 2.27793397399067280805995364883, 3.25573402263557505245686635144, 4.49953796443089226654373359208, 6.12026712749926761754033053875, 7.19156332721071557759182944943, 8.180195219483340050366861115161, 8.756580632829602207354766995578, 9.866063339378579790894242287643, 10.54010086325438131626714470570