L(s) = 1 | − 2-s + (0.396 − 1.68i)3-s + 4-s − 2.37i·5-s + (−0.396 + 1.68i)6-s + (−0.792 + 2.52i)7-s − 8-s + (−2.68 − 1.33i)9-s + 2.37i·10-s − 5.74·11-s + (0.396 − 1.68i)12-s + (3.46 − i)13-s + (0.792 − 2.52i)14-s + (−4 − 0.939i)15-s + 16-s − 2.52·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.228 − 0.973i)3-s + 0.5·4-s − 1.06i·5-s + (−0.161 + 0.688i)6-s + (−0.299 + 0.954i)7-s − 0.353·8-s + (−0.895 − 0.445i)9-s + 0.750i·10-s − 1.73·11-s + (0.114 − 0.486i)12-s + (0.960 − 0.277i)13-s + (0.211 − 0.674i)14-s + (−1.03 − 0.242i)15-s + 0.250·16-s − 0.612·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0519604 + 0.407188i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0519604 + 0.407188i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.396 + 1.68i)T \) |
| 7 | \( 1 + (0.792 - 2.52i)T \) |
| 13 | \( 1 + (-3.46 + i)T \) |
good | 5 | \( 1 + 2.37iT - 5T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 17 | \( 1 + 2.52T + 17T^{2} \) |
| 19 | \( 1 + 7.57T + 19T^{2} \) |
| 23 | \( 1 + 6.78iT - 23T^{2} \) |
| 29 | \( 1 - 7.57iT - 29T^{2} \) |
| 31 | \( 1 - 5.84T + 31T^{2} \) |
| 37 | \( 1 + 4.90iT - 37T^{2} \) |
| 41 | \( 1 + 2.62iT - 41T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 - 4.62iT - 47T^{2} \) |
| 53 | \( 1 - 3.16iT - 53T^{2} \) |
| 59 | \( 1 + 8.74iT - 59T^{2} \) |
| 61 | \( 1 + 3.74iT - 61T^{2} \) |
| 67 | \( 1 + 2.67iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 3.61T + 73T^{2} \) |
| 79 | \( 1 + 9.37T + 79T^{2} \) |
| 83 | \( 1 + 4.74iT - 83T^{2} \) |
| 89 | \( 1 + 10iT - 89T^{2} \) |
| 97 | \( 1 - 7.42T + 97T^{2} \) |
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
−10.37470122271089255011943822903, −8.922533207385941947531764497491, −8.568648675280449973130159527757, −8.059128488771197434911033200021, −6.70252625517887522507509840441, −5.93018510992212937303659922940, −4.86434962628243197554175576965, −2.90699640752344210620984506210, −1.94522496819574800944580624884, −0.26500124053925320592168741409,
2.41651588967210866406652695902, 3.40903143010740773321449203538, 4.50382230276382722376555395331, 5.96414690619542496315461426954, 6.84539190962808135822471006586, 7.935660837436565935698352570810, 8.562084242192857338972214078851, 9.898618215111414364005957303467, 10.29923741793137044572738804263, 10.91596310155268621166888602029