| L(s) = 1 | − 2·7-s + 4·11-s + 13-s + 6·19-s + 4·23-s − 8·29-s + 2·31-s − 6·37-s + 6·41-s − 8·43-s + 8·47-s − 3·49-s − 12·53-s − 4·59-s + 10·61-s − 2·67-s + 16·71-s − 14·73-s − 8·77-s + 4·79-s − 12·83-s − 6·89-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.20·11-s + 0.277·13-s + 1.37·19-s + 0.834·23-s − 1.48·29-s + 0.359·31-s − 0.986·37-s + 0.937·41-s − 1.21·43-s + 1.16·47-s − 3/7·49-s − 1.64·53-s − 0.520·59-s + 1.28·61-s − 0.244·67-s + 1.89·71-s − 1.63·73-s − 0.911·77-s + 0.450·79-s − 1.31·83-s − 0.635·89-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.72023166414783, −14.41085157486591, −13.77240742889816, −13.37372283193375, −12.78854495718139, −12.29450208042835, −11.77889534364189, −11.20923323440051, −10.88469452396301, −9.967852727930208, −9.600135600744061, −9.205009363564988, −8.679309287656927, −7.970424686328419, −7.275167495019942, −6.889185037942924, −6.332484050258086, −5.702464262565376, −5.179359443256891, −4.414609258260958, −3.639723531342638, −3.373810192196172, −2.606525518312358, −1.601784464658822, −1.066018047463074, 0,
1.066018047463074, 1.601784464658822, 2.606525518312358, 3.373810192196172, 3.639723531342638, 4.414609258260958, 5.179359443256891, 5.702464262565376, 6.332484050258086, 6.889185037942924, 7.275167495019942, 7.970424686328419, 8.679309287656927, 9.205009363564988, 9.600135600744061, 9.967852727930208, 10.88469452396301, 11.20923323440051, 11.77889534364189, 12.29450208042835, 12.78854495718139, 13.37372283193375, 13.77240742889816, 14.41085157486591, 14.72023166414783
