| L(s) = 1 | − 3·3-s − 4·5-s − 2·7-s + 6·9-s − 5·11-s + 2·13-s + 12·15-s − 6·17-s − 2·19-s + 6·21-s − 6·23-s + 5·25-s − 9·27-s + 2·29-s − 4·31-s + 15·33-s + 8·35-s − 16·37-s − 6·39-s − 41-s − 7·43-s − 24·45-s + 2·47-s + 7·49-s + 18·51-s − 8·53-s + 20·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.78·5-s − 0.755·7-s + 2·9-s − 1.50·11-s + 0.554·13-s + 3.09·15-s − 1.45·17-s − 0.458·19-s + 1.30·21-s − 1.25·23-s + 25-s − 1.73·27-s + 0.371·29-s − 0.718·31-s + 2.61·33-s + 1.35·35-s − 2.63·37-s − 0.960·39-s − 0.156·41-s − 1.06·43-s − 3.57·45-s + 0.291·47-s + 49-s + 2.52·51-s − 1.09·53-s + 2.69·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−11.59877513113809411037446217947, −11.12079660481693041266681147288, −10.73279830984871759150534901639, −10.40180227503275119799615289251, −10.05711842340712488234646729413, −9.214144988788280623398567386581, −8.471468801425551706409650535733, −8.299018976219332246012939701247, −7.37874357245076990176121131975, −7.29739171260601454470072494719, −6.55508742477961674921517008471, −6.17706241706222524722509405326, −5.49538195912284908170831401640, −4.96200882829849467334325646005, −4.35421603453266661837790422805, −3.87080605634021113619894534701, −3.20240967842739911059694004718, −1.95569520263527881773828170727, 0, 0,
1.95569520263527881773828170727, 3.20240967842739911059694004718, 3.87080605634021113619894534701, 4.35421603453266661837790422805, 4.96200882829849467334325646005, 5.49538195912284908170831401640, 6.17706241706222524722509405326, 6.55508742477961674921517008471, 7.29739171260601454470072494719, 7.37874357245076990176121131975, 8.299018976219332246012939701247, 8.471468801425551706409650535733, 9.214144988788280623398567386581, 10.05711842340712488234646729413, 10.40180227503275119799615289251, 10.73279830984871759150534901639, 11.12079660481693041266681147288, 11.59877513113809411037446217947
