| L(s) = 1 | − 2-s − 4-s − 5-s − 4·7-s + 3·8-s + 10-s − 4·11-s + 13-s + 4·14-s − 16-s + 20-s + 4·22-s + 25-s − 26-s + 4·28-s + 6·29-s + 4·31-s − 5·32-s + 4·35-s + 10·37-s − 3·40-s − 2·41-s + 4·43-s + 4·44-s + 9·49-s − 50-s − 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.51·7-s + 1.06·8-s + 0.316·10-s − 1.20·11-s + 0.277·13-s + 1.06·14-s − 1/4·16-s + 0.223·20-s + 0.852·22-s + 1/5·25-s − 0.196·26-s + 0.755·28-s + 1.11·29-s + 0.718·31-s − 0.883·32-s + 0.676·35-s + 1.64·37-s − 0.474·40-s − 0.312·41-s + 0.609·43-s + 0.603·44-s + 9/7·49-s − 0.141·50-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−13.32952752119843, −12.97207554668364, −12.74529450727361, −12.01494053035978, −11.58903373421512, −10.88598983852053, −10.41107545928116, −10.02956142690666, −9.828573777289567, −9.021883714012971, −8.812763793382008, −8.210968546105615, −7.697582517693753, −7.391904430843847, −6.663765310926522, −6.239625559915932, −5.657674251107463, −4.994790505828295, −4.532511920711608, −3.879262148959287, −3.441307043684423, −2.652054584708390, −2.399385906703144, −1.138530386646095, −0.6595637219422404, 0,
0.6595637219422404, 1.138530386646095, 2.399385906703144, 2.652054584708390, 3.441307043684423, 3.879262148959287, 4.532511920711608, 4.994790505828295, 5.657674251107463, 6.239625559915932, 6.663765310926522, 7.391904430843847, 7.697582517693753, 8.210968546105615, 8.812763793382008, 9.021883714012971, 9.828573777289567, 10.02956142690666, 10.41107545928116, 10.88598983852053, 11.58903373421512, 12.01494053035978, 12.74529450727361, 12.97207554668364, 13.32952752119843
