| L(s) = 1 | − 5-s + 7-s + 6·13-s + 3·17-s − 5·19-s − 2·23-s + 25-s − 5·29-s − 5·31-s − 35-s − 37-s − 2·41-s + 12·43-s − 2·47-s − 6·49-s + 13·53-s + 2·59-s − 61-s − 6·65-s − 16·67-s + 15·71-s − 10·73-s + 2·79-s + 14·83-s − 3·85-s − 9·89-s + 6·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.66·13-s + 0.727·17-s − 1.14·19-s − 0.417·23-s + 1/5·25-s − 0.928·29-s − 0.898·31-s − 0.169·35-s − 0.164·37-s − 0.312·41-s + 1.82·43-s − 0.291·47-s − 6/7·49-s + 1.78·53-s + 0.260·59-s − 0.128·61-s − 0.744·65-s − 1.95·67-s + 1.78·71-s − 1.17·73-s + 0.225·79-s + 1.53·83-s − 0.325·85-s − 0.953·89-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−14.06705335051340, −13.75549852352639, −13.00316144732493, −12.81605568769344, −12.14827440928610, −11.62342814014048, −11.15513453767838, −10.71207796138373, −10.39536119440419, −9.604354741592687, −8.979766345752114, −8.672545263630735, −8.069511104519693, −7.684557779120058, −7.069808259579961, −6.451270853823614, −5.844657866525187, −5.542926789974398, −4.728360610364972, −3.966936195706456, −3.839739066400467, −3.103613559812108, −2.249688725792683, −1.608664874873281, −0.9441137457846837, 0,
0.9441137457846837, 1.608664874873281, 2.249688725792683, 3.103613559812108, 3.839739066400467, 3.966936195706456, 4.728360610364972, 5.542926789974398, 5.844657866525187, 6.451270853823614, 7.069808259579961, 7.684557779120058, 8.069511104519693, 8.672545263630735, 8.979766345752114, 9.604354741592687, 10.39536119440419, 10.71207796138373, 11.15513453767838, 11.62342814014048, 12.14827440928610, 12.81605568769344, 13.00316144732493, 13.75549852352639, 14.06705335051340
