| L(s) = 1 | − 243·3-s − 1.17e4·5-s + 5.00e4·7-s + 5.90e4·9-s + 5.31e5·11-s + 1.33e6·13-s + 2.85e6·15-s − 5.10e6·17-s − 2.90e6·19-s − 1.21e7·21-s − 3.05e7·23-s + 8.87e7·25-s − 1.43e7·27-s − 7.70e7·29-s + 2.39e8·31-s − 1.29e8·33-s − 5.86e8·35-s − 7.85e8·37-s − 3.23e8·39-s + 4.11e8·41-s − 3.51e8·43-s − 6.92e8·45-s − 9.58e7·47-s + 5.23e8·49-s + 1.24e9·51-s − 1.46e9·53-s − 6.23e9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.67·5-s + 1.12·7-s + 1/3·9-s + 0.994·11-s + 0.995·13-s + 0.969·15-s − 0.872·17-s − 0.268·19-s − 0.649·21-s − 0.991·23-s + 1.81·25-s − 0.192·27-s − 0.697·29-s + 1.50·31-s − 0.574·33-s − 1.88·35-s − 1.86·37-s − 0.574·39-s + 0.554·41-s − 0.364·43-s − 0.559·45-s − 0.0609·47-s + 0.264·49-s + 0.503·51-s − 0.481·53-s − 1.67·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{5} T \) |
| good | 5 | \( 1 + 2346 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 7144 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 531420 T + p^{11} T^{2} \) |
| 13 | \( 1 - 1332566 T + p^{11} T^{2} \) |
| 17 | \( 1 + 5109678 T + p^{11} T^{2} \) |
| 19 | \( 1 + 2901404 T + p^{11} T^{2} \) |
| 23 | \( 1 + 30597000 T + p^{11} T^{2} \) |
| 29 | \( 1 + 77006634 T + p^{11} T^{2} \) |
| 31 | \( 1 - 239418352 T + p^{11} T^{2} \) |
| 37 | \( 1 + 785041666 T + p^{11} T^{2} \) |
| 41 | \( 1 - 411252954 T + p^{11} T^{2} \) |
| 43 | \( 1 + 351233348 T + p^{11} T^{2} \) |
| 47 | \( 1 + 95821680 T + p^{11} T^{2} \) |
| 53 | \( 1 + 1465857378 T + p^{11} T^{2} \) |
| 59 | \( 1 + 5621152020 T + p^{11} T^{2} \) |
| 61 | \( 1 + 10473587770 T + p^{11} T^{2} \) |
| 67 | \( 1 + 4515307532 T + p^{11} T^{2} \) |
| 71 | \( 1 - 8509579560 T + p^{11} T^{2} \) |
| 73 | \( 1 - 2012496986 T + p^{11} T^{2} \) |
| 79 | \( 1 - 22238409568 T + p^{11} T^{2} \) |
| 83 | \( 1 + 6328647516 T + p^{11} T^{2} \) |
| 89 | \( 1 + 50123706678 T + p^{11} T^{2} \) |
| 97 | \( 1 - 94805961314 T + p^{11} T^{2} \) |
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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
−12.22566077993171707773690531591, −11.52362206646557929598278960329, −10.80597526673119359928489536389, −8.737579225378019868605587474175, −7.80517034377636400038760707307, −6.46576812489672518189068779381, −4.64826417587334717435397598423, −3.79084561638491104152401773494, −1.42308576600579533970043602578, 0,
1.42308576600579533970043602578, 3.79084561638491104152401773494, 4.64826417587334717435397598423, 6.46576812489672518189068779381, 7.80517034377636400038760707307, 8.737579225378019868605587474175, 10.80597526673119359928489536389, 11.52362206646557929598278960329, 12.22566077993171707773690531591
