L(s) = 1 | + (−0.385 − 0.667i)5-s + (−2.20 − 1.46i)7-s + (4.68 + 2.70i)11-s + 5.28i·13-s + (2.85 − 4.94i)17-s + (0.535 − 0.309i)19-s + (5.83 − 3.37i)23-s + (2.20 − 3.81i)25-s − 8.07i·29-s + (0.502 + 0.290i)31-s + (−0.129 + 2.03i)35-s + (−4.53 − 7.86i)37-s + 8.59·41-s + 3.40·43-s + (−0.385 − 0.667i)47-s + ⋯ |
L(s) = 1 | + (−0.172 − 0.298i)5-s + (−0.832 − 0.553i)7-s + (1.41 + 0.814i)11-s + 1.46i·13-s + (0.692 − 1.19i)17-s + (0.122 − 0.0709i)19-s + (1.21 − 0.702i)23-s + (0.440 − 0.763i)25-s − 1.49i·29-s + (0.0902 + 0.0521i)31-s + (−0.0218 + 0.344i)35-s + (−0.746 − 1.29i)37-s + 1.34·41-s + 0.519·43-s + (−0.0562 − 0.0973i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44024 - 0.337394i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44024 - 0.337394i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.20 + 1.46i)T \) |
good | 5 | \( 1 + (0.385 + 0.667i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.68 - 2.70i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.28iT - 13T^{2} \) |
| 17 | \( 1 + (-2.85 + 4.94i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.535 + 0.309i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.83 + 3.37i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 8.07iT - 29T^{2} \) |
| 31 | \( 1 + (-0.502 - 0.290i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.53 + 7.86i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.59T + 41T^{2} \) |
| 43 | \( 1 - 3.40T + 43T^{2} \) |
| 47 | \( 1 + (0.385 + 0.667i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.63 - 3.83i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.89 - 11.9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.57 - 2.06i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.77 - 11.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.25iT - 71T^{2} \) |
| 73 | \( 1 + (-1.96 - 1.13i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.03 - 6.98i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.85T + 83T^{2} \) |
| 89 | \( 1 + (2.59 + 4.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 14.0iT - 97T^{2} \) |
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
−10.11520131839670974464627439272, −9.281823710162567244250164600871, −8.964242798057609718179219813756, −7.35393534533121030432409756423, −6.95763970611136352789003924869, −6.01982930369405296743315298377, −4.51447752292528500597526972493, −4.04923412599405215337480570000, −2.59877565754128017419934403976, −0.967623289355127524960642564385,
1.21405868163707343823062386748, 3.24323527392908842849655263182, 3.44018328760876710518272396583, 5.21181262387387308085775039683, 6.02889427132190634831129697253, 6.79607546553438792485238226181, 7.86783232509646237856403709221, 8.822579681279055905419283342690, 9.428787144776210199247561791705, 10.51478292199814955179882634083