| L(s) = 1 | − 3-s + 9-s − 2·11-s + 2·13-s + 4·17-s − 8·23-s − 27-s − 2·31-s + 2·33-s + 8·37-s − 2·39-s − 2·41-s − 2·43-s − 10·47-s − 4·51-s − 2·53-s − 4·59-s + 10·61-s + 2·67-s + 8·69-s − 12·71-s − 10·73-s + 16·79-s + 81-s + 16·83-s + 14·89-s + 2·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.970·17-s − 1.66·23-s − 0.192·27-s − 0.359·31-s + 0.348·33-s + 1.31·37-s − 0.320·39-s − 0.312·41-s − 0.304·43-s − 1.45·47-s − 0.560·51-s − 0.274·53-s − 0.520·59-s + 1.28·61-s + 0.244·67-s + 0.963·69-s − 1.42·71-s − 1.17·73-s + 1.80·79-s + 1/9·81-s + 1.75·83-s + 1.48·89-s + 0.207·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.613436832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.613436832\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.83837059518807, −12.47583625898524, −11.89417666464088, −11.52932036848549, −11.18813249392469, −10.44011019746679, −10.22443563503375, −9.796241018072523, −9.261307453299981, −8.619650577992081, −8.120501159959119, −7.663074594868605, −7.404824562574366, −6.486885051923199, −6.194238127401514, −5.809601596023573, −5.182586197742359, −4.754788828065466, −4.177258877745486, −3.474807742399971, −3.222112079453033, −2.226844325810907, −1.856650997964227, −1.034073764739358, −0.4082005529925504,
0.4082005529925504, 1.034073764739358, 1.856650997964227, 2.226844325810907, 3.222112079453033, 3.474807742399971, 4.177258877745486, 4.754788828065466, 5.182586197742359, 5.809601596023573, 6.194238127401514, 6.486885051923199, 7.404824562574366, 7.663074594868605, 8.120501159959119, 8.619650577992081, 9.261307453299981, 9.796241018072523, 10.22443563503375, 10.44011019746679, 11.18813249392469, 11.52932036848549, 11.89417666464088, 12.47583625898524, 12.83837059518807
