| L(s) = 1 | − 3·5-s + 7-s − 3·11-s − 13-s − 12·17-s + 8·19-s − 3·23-s + 5·25-s − 3·29-s − 5·31-s − 3·35-s − 4·37-s + 3·41-s − 43-s − 9·47-s + 7·49-s − 12·53-s + 9·55-s + 3·59-s − 13·61-s + 3·65-s − 7·67-s + 24·71-s − 20·73-s − 3·77-s − 11·79-s + 9·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s − 0.904·11-s − 0.277·13-s − 2.91·17-s + 1.83·19-s − 0.625·23-s + 25-s − 0.557·29-s − 0.898·31-s − 0.507·35-s − 0.657·37-s + 0.468·41-s − 0.152·43-s − 1.31·47-s + 49-s − 1.64·53-s + 1.21·55-s + 0.390·59-s − 1.66·61-s + 0.372·65-s − 0.855·67-s + 2.84·71-s − 2.34·73-s − 0.341·77-s − 1.23·79-s + 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.916202765149935336340970175563, −8.896638656111439619872880877728, −8.196861099471919525723328454638, −7.932506839450750136561253281174, −7.51949580913777253593583752806, −7.33797396553104296452270534206, −6.77139654011188818892121184951, −6.49133796553145479714456176018, −5.83269159480779106557759574530, −5.33524603553496783036987217920, −4.91356896588412488798167838108, −4.60904772100616062882817729302, −4.07185540848452660517665422806, −3.75082639121020168292577485544, −3.06075514713325963722460260782, −2.67959578799257357047500303303, −2.00957822122610717881840022745, −1.38629320182771795568895917193, 0, 0,
1.38629320182771795568895917193, 2.00957822122610717881840022745, 2.67959578799257357047500303303, 3.06075514713325963722460260782, 3.75082639121020168292577485544, 4.07185540848452660517665422806, 4.60904772100616062882817729302, 4.91356896588412488798167838108, 5.33524603553496783036987217920, 5.83269159480779106557759574530, 6.49133796553145479714456176018, 6.77139654011188818892121184951, 7.33797396553104296452270534206, 7.51949580913777253593583752806, 7.932506839450750136561253281174, 8.196861099471919525723328454638, 8.896638656111439619872880877728, 8.916202765149935336340970175563
