| L(s) = 1 | + 2-s − 4-s + 5-s − 3·8-s + 10-s + 11-s + 2·13-s − 16-s + 2·17-s − 4·19-s − 20-s + 22-s + 25-s + 2·26-s − 6·29-s + 5·32-s + 2·34-s + 6·37-s − 4·38-s − 3·40-s − 6·41-s − 4·43-s − 44-s + 50-s − 2·52-s + 2·53-s + 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 + 0.301·11-s + 0.554·13-s − 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.392·26-s − 1.11·29-s + 0.883·32-s + 0.342·34-s + 0.986·37-s − 0.648·38-s − 0.474·40-s − 0.937·41-s − 0.609·43-s − 0.150·44-s + 0.141·50-s − 0.277·52-s + 0.274·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24255 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24255 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.44301139145692, −14.85883021124367, −14.71634900872222, −13.96636136180827, −13.48286882340618, −13.13534635323620, −12.56743477386050, −12.06564799934034, −11.40274421882807, −10.87705183513647, −10.13208325339697, −9.624038254671315, −9.107740553212278, −8.494376326720685, −8.046555416003350, −7.138442573000857, −6.448992519552913, −5.976442387871612, −5.399931806600598, −4.811350875169811, −4.054827212154791, −3.626847885207321, −2.855532204544586, −2.011746024535311, −1.110400014313263, 0,
1.110400014313263, 2.011746024535311, 2.855532204544586, 3.626847885207321, 4.054827212154791, 4.811350875169811, 5.399931806600598, 5.976442387871612, 6.448992519552913, 7.138442573000857, 8.046555416003350, 8.494376326720685, 9.107740553212278, 9.624038254671315, 10.13208325339697, 10.87705183513647, 11.40274421882807, 12.06564799934034, 12.56743477386050, 13.13534635323620, 13.48286882340618, 13.96636136180827, 14.71634900872222, 14.85883021124367, 15.44301139145692
