| L(s) = 1 | − 3-s + 9-s + 4·11-s − 2·13-s + 2·17-s + 4·19-s − 8·23-s − 27-s − 2·29-s − 4·33-s − 6·37-s + 2·39-s + 6·41-s − 4·43-s − 2·51-s + 10·53-s − 4·57-s + 12·59-s − 14·61-s − 12·67-s + 8·69-s + 8·71-s + 10·73-s − 16·79-s + 81-s + 12·83-s + 2·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s − 0.192·27-s − 0.371·29-s − 0.696·33-s − 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.280·51-s + 1.37·53-s − 0.529·57-s + 1.56·59-s − 1.79·61-s − 1.46·67-s + 0.963·69-s + 0.949·71-s + 1.17·73-s − 1.80·79-s + 1/9·81-s + 1.31·83-s + 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.44169857968014, −14.16272931916142, −13.65638235177305, −13.04311550150385, −12.28999069939731, −12.04283784724384, −11.71370177323753, −11.12698684876810, −10.45107934604543, −9.933483873363820, −9.618509570912115, −8.958097425611134, −8.430230732740851, −7.599514409074612, −7.366361887848255, −6.638475371156640, −6.135394612559591, −5.583405686759339, −5.090588820514485, −4.303510380346374, −3.852118335938063, −3.249542338992923, −2.318340338816881, −1.630228157920594, −0.9398653664482444, 0,
0.9398653664482444, 1.630228157920594, 2.318340338816881, 3.249542338992923, 3.852118335938063, 4.303510380346374, 5.090588820514485, 5.583405686759339, 6.135394612559591, 6.638475371156640, 7.366361887848255, 7.599514409074612, 8.430230732740851, 8.958097425611134, 9.618509570912115, 9.933483873363820, 10.45107934604543, 11.12698684876810, 11.71370177323753, 12.04283784724384, 12.28999069939731, 13.04311550150385, 13.65638235177305, 14.16272931916142, 14.44169857968014
