| L(s) = 1 | + 3-s − 2·5-s + 5·7-s + 6·11-s − 6·13-s − 2·15-s − 4·17-s + 5·19-s + 5·21-s + 4·23-s + 5·25-s − 27-s − 8·29-s − 7·31-s + 6·33-s − 10·35-s + 9·37-s − 6·39-s − 4·41-s − 2·43-s − 2·47-s + 18·49-s − 4·51-s − 8·53-s − 12·55-s + 5·57-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.88·7-s + 1.80·11-s − 1.66·13-s − 0.516·15-s − 0.970·17-s + 1.14·19-s + 1.09·21-s + 0.834·23-s + 25-s − 0.192·27-s − 1.48·29-s − 1.25·31-s + 1.04·33-s − 1.69·35-s + 1.47·37-s − 0.960·39-s − 0.624·41-s − 0.304·43-s − 0.291·47-s + 18/7·49-s − 0.560·51-s − 1.09·53-s − 1.61·55-s + 0.662·57-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.561762216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.561762216\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 8 T + 11 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−12.93998168163479237060150960111, −12.49647059581523935845655071106, −11.80810672278820619933441987182, −11.63213957703065132972287836513, −11.10923750331065684197624462419, −10.93984127357960550447846175547, −9.865810713128648151345970312920, −9.267857396078675292804220194486, −9.066908169917119266659675428788, −8.455168944411838828307103830164, −7.74939842826450915271496415580, −7.46661102487355113936002959526, −7.06202613067146614143981319815, −6.22004389258745725172411486743, −4.99206537175416333524925458485, −4.98644188984549650667574953911, −4.10700387459099160093599550272, −3.51752340962738967894359338796, −2.39332483157576102990205156124, −1.45509825848733654655920391659,
1.45509825848733654655920391659, 2.39332483157576102990205156124, 3.51752340962738967894359338796, 4.10700387459099160093599550272, 4.98644188984549650667574953911, 4.99206537175416333524925458485, 6.22004389258745725172411486743, 7.06202613067146614143981319815, 7.46661102487355113936002959526, 7.74939842826450915271496415580, 8.455168944411838828307103830164, 9.066908169917119266659675428788, 9.267857396078675292804220194486, 9.865810713128648151345970312920, 10.93984127357960550447846175547, 11.10923750331065684197624462419, 11.63213957703065132972287836513, 11.80810672278820619933441987182, 12.49647059581523935845655071106, 12.93998168163479237060150960111
