| L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s + 10-s + 4·11-s + 2·13-s − 16-s − 2·17-s + 20-s − 4·22-s + 25-s − 2·26-s − 2·29-s − 5·32-s + 2·34-s + 10·37-s − 3·40-s + 10·41-s + 4·43-s − 4·44-s − 8·47-s − 7·49-s − 50-s − 2·52-s − 10·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s + 0.316·10-s + 1.20·11-s + 0.554·13-s − 1/4·16-s − 0.485·17-s + 0.223·20-s − 0.852·22-s + 1/5·25-s − 0.392·26-s − 0.371·29-s − 0.883·32-s + 0.342·34-s + 1.64·37-s − 0.474·40-s + 1.56·41-s + 0.609·43-s − 0.603·44-s − 1.16·47-s − 49-s − 0.141·50-s − 0.277·52-s − 1.37·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−16.34505631804697, −15.87992623199518, −15.06783458472703, −14.50862692030166, −14.14762779108103, −13.39178409409869, −12.92145001425524, −12.37124121263743, −11.52774869509717, −11.12487971840822, −10.68440398566746, −9.680093478063023, −9.437954228187323, −8.896447809747983, −8.232011963291614, −7.770712806775348, −7.115240131362891, −6.341451675560947, −5.836793908336158, −4.721092477623742, −4.331956660361172, −3.724926864673544, −2.853806729199446, −1.663419210706286, −1.050293617830049, 0,
1.050293617830049, 1.663419210706286, 2.853806729199446, 3.724926864673544, 4.331956660361172, 4.721092477623742, 5.836793908336158, 6.341451675560947, 7.115240131362891, 7.770712806775348, 8.232011963291614, 8.896447809747983, 9.437954228187323, 9.680093478063023, 10.68440398566746, 11.12487971840822, 11.52774869509717, 12.37124121263743, 12.92145001425524, 13.39178409409869, 14.14762779108103, 14.50862692030166, 15.06783458472703, 15.87992623199518, 16.34505631804697
