| L(s) = 1 | + 2·2-s + 2·4-s + 4·5-s − 9-s + 8·10-s − 2·11-s − 6·13-s − 4·16-s − 2·18-s − 4·19-s + 8·20-s − 4·22-s + 12·23-s + 8·25-s − 12·26-s + 20·29-s − 12·31-s − 8·32-s − 2·36-s + 6·37-s − 8·38-s − 8·41-s − 8·43-s − 4·44-s − 4·45-s + 24·46-s − 10·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.78·5-s − 1/3·9-s + 2.52·10-s − 0.603·11-s − 1.66·13-s − 16-s − 0.471·18-s − 0.917·19-s + 1.78·20-s − 0.852·22-s + 2.50·23-s + 8/5·25-s − 2.35·26-s + 3.71·29-s − 2.15·31-s − 1.41·32-s − 1/3·36-s + 0.986·37-s − 1.29·38-s − 1.24·41-s − 1.21·43-s − 0.603·44-s − 0.596·45-s + 3.53·46-s − 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.752503430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.752503430\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−13.11667038927838868674805007168, −12.78984925491994985138214337046, −12.59324643773461368391088418112, −11.87722591546510555509764443414, −11.29162216668021452497130329084, −10.72671240281135793083914490681, −10.11419295141218122243579561841, −9.867486814398814440346734390473, −9.031042526724114408248042140599, −8.819766817104925505631317767067, −7.902900335850924252865934535119, −7.01539617270518702129499545393, −6.51021014914383023851942059687, −6.23394179950514215914425328607, −5.20439695068005364171344142967, −5.00732782805905316363820054731, −4.69928451732084845466623018503, −3.14663386142351878046677138441, −2.79546562205370131907837562595, −1.96881106737587450189060742595,
1.96881106737587450189060742595, 2.79546562205370131907837562595, 3.14663386142351878046677138441, 4.69928451732084845466623018503, 5.00732782805905316363820054731, 5.20439695068005364171344142967, 6.23394179950514215914425328607, 6.51021014914383023851942059687, 7.01539617270518702129499545393, 7.902900335850924252865934535119, 8.819766817104925505631317767067, 9.031042526724114408248042140599, 9.867486814398814440346734390473, 10.11419295141218122243579561841, 10.72671240281135793083914490681, 11.29162216668021452497130329084, 11.87722591546510555509764443414, 12.59324643773461368391088418112, 12.78984925491994985138214337046, 13.11667038927838868674805007168
