| L(s) = 1 | + 5-s + 4·7-s + 4·11-s − 13-s − 6·17-s + 4·23-s + 25-s − 6·29-s − 8·31-s + 4·35-s + 2·37-s − 10·41-s + 4·43-s − 8·47-s + 9·49-s − 2·53-s + 4·55-s + 4·59-s − 14·61-s − 65-s + 12·67-s + 8·71-s − 10·73-s + 16·77-s − 4·83-s − 6·85-s − 10·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 1.20·11-s − 0.277·13-s − 1.45·17-s + 0.834·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.676·35-s + 0.328·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.274·53-s + 0.539·55-s + 0.520·59-s − 1.79·61-s − 0.124·65-s + 1.46·67-s + 0.949·71-s − 1.17·73-s + 1.82·77-s − 0.439·83-s − 0.650·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−14.91686921399812, −14.74332168967363, −14.07911614599567, −13.72447331892035, −12.96451414804188, −12.67430527947973, −11.66284462151861, −11.55684718454248, −10.92703796354613, −10.63146809094561, −9.614391973643196, −9.266066703654967, −8.764369281650985, −8.238540708334308, −7.561938977297054, −6.907538056080750, −6.578376872761616, −5.675171490661097, −5.188621148471912, −4.585617617970808, −4.075796291934814, −3.321956528613692, −2.326594826847864, −1.741452146632938, −1.298866783328736, 0,
1.298866783328736, 1.741452146632938, 2.326594826847864, 3.321956528613692, 4.075796291934814, 4.585617617970808, 5.188621148471912, 5.675171490661097, 6.578376872761616, 6.907538056080750, 7.561938977297054, 8.238540708334308, 8.764369281650985, 9.266066703654967, 9.614391973643196, 10.63146809094561, 10.92703796354613, 11.55684718454248, 11.66284462151861, 12.67430527947973, 12.96451414804188, 13.72447331892035, 14.07911614599567, 14.74332168967363, 14.91686921399812
