L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.881 + 2.71i)5-s + (1.30 + 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.927 + 2.85i)9-s + 2.85·10-s + (−3.04 + 1.31i)11-s + (−2 + 6.15i)13-s + (1.30 − 0.951i)14-s + (0.309 + 0.951i)16-s + (−2.19 − 6.74i)17-s + (2.42 + 1.76i)18-s + (−0.809 + 0.587i)19-s + (0.881 − 2.71i)20-s + ⋯ |
L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (0.394 + 1.21i)5-s + (0.494 + 0.359i)7-s + (−0.286 + 0.207i)8-s + (−0.309 + 0.951i)9-s + 0.902·10-s + (−0.918 + 0.396i)11-s + (−0.554 + 1.70i)13-s + (0.349 − 0.254i)14-s + (0.0772 + 0.237i)16-s + (−0.531 − 1.63i)17-s + (0.572 + 0.415i)18-s + (−0.185 + 0.134i)19-s + (0.197 − 0.606i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29686 + 0.523545i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29686 + 0.523545i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 + (3.04 - 1.31i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
good | 3 | \( 1 + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (-0.881 - 2.71i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (-1.30 - 0.951i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (2 - 6.15i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (2.19 + 6.74i)T + (-13.7 + 9.99i)T^{2} \) |
| 23 | \( 1 - 7.85T + 23T^{2} \) |
| 29 | \( 1 + (-2.61 - 1.90i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.38 + 7.33i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.85 - 2.80i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.23 - 0.898i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 0.618T + 43T^{2} \) |
| 47 | \( 1 + (-7.59 + 5.51i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.618 + 1.90i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-8.09 - 5.87i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.336 + 1.03i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + (0.145 + 0.449i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.381 + 0.277i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (3.70 - 11.4i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.881 + 2.71i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 8.18T + 89T^{2} \) |
| 97 | \( 1 + (5.09 - 15.6i)T + (-78.4 - 57.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.35126731987160974985805176734, −10.58069397604981323884425522376, −9.723369325123647609238436270736, −8.804307239983112982996551053528, −7.46220679469244254824410716168, −6.71881898000411231284983709597, −5.26546451173855260370565130317, −4.56037786507942297073302732816, −2.66694860294496584681334085296, −2.28573191949558345693742648555,
0.860856831281497763554662376782, 3.05709982234276060185523615154, 4.52951266680804488479275254185, 5.35481768084013896823237778449, 6.13108460847911043587454809768, 7.49627530444988714730757975594, 8.436712610083848133007109235980, 8.883868377391657285815345371748, 10.16526284074638926470062066308, 10.97438948930478800963000684739