| L(s) = 1 | − 7-s + 4·11-s − 6·13-s − 6·17-s − 4·19-s + 8·23-s − 10·29-s − 4·31-s + 6·37-s − 6·41-s + 4·43-s + 12·47-s + 49-s + 6·53-s + 4·59-s − 2·61-s + 4·67-s + 2·73-s − 4·77-s + 8·79-s + 12·83-s − 14·89-s + 6·91-s − 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.20·11-s − 1.66·13-s − 1.45·17-s − 0.917·19-s + 1.66·23-s − 1.85·29-s − 0.718·31-s + 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s − 0.256·61-s + 0.488·67-s + 0.234·73-s − 0.455·77-s + 0.900·79-s + 1.31·83-s − 1.48·89-s + 0.628·91-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.92913106007220, −14.40645983998031, −13.67524689891010, −13.16972225170502, −12.72822731731999, −12.30908724109633, −11.66683538299330, −11.10466035898002, −10.81530634477457, −10.03171387941150, −9.441850807360975, −8.983161994310501, −8.858689528681222, −7.791786360174712, −7.247718738785021, −6.842275249908808, −6.414609009722020, −5.576442928252832, −5.085098897720196, −4.264783085526208, −4.020736342850491, −3.136588852519621, −2.327449665078389, −1.982661497358179, −0.8446966422730738, 0,
0.8446966422730738, 1.982661497358179, 2.327449665078389, 3.136588852519621, 4.020736342850491, 4.264783085526208, 5.085098897720196, 5.576442928252832, 6.414609009722020, 6.842275249908808, 7.247718738785021, 7.791786360174712, 8.858689528681222, 8.983161994310501, 9.441850807360975, 10.03171387941150, 10.81530634477457, 11.10466035898002, 11.66683538299330, 12.30908724109633, 12.72822731731999, 13.16972225170502, 13.67524689891010, 14.40645983998031, 14.92913106007220
