| L(s) = 1 | + 46·3-s − 160·5-s − 343·7-s − 71·9-s − 6.84e3·11-s − 2.90e3·13-s − 7.36e3·15-s + 1.65e4·17-s − 6.71e3·19-s − 1.57e4·21-s − 976·23-s − 5.25e4·25-s − 1.03e5·27-s − 6.16e4·29-s − 6.92e4·31-s − 3.14e5·33-s + 5.48e4·35-s − 5.33e5·37-s − 1.33e5·39-s + 1.83e5·41-s + 9.66e5·43-s + 1.13e4·45-s − 1.90e5·47-s + 1.17e5·49-s + 7.62e5·51-s − 7.85e5·53-s + 1.09e6·55-s + ⋯ |
| L(s) = 1 | + 0.983·3-s − 0.572·5-s − 0.377·7-s − 0.0324·9-s − 1.54·11-s − 0.366·13-s − 0.563·15-s + 0.817·17-s − 0.224·19-s − 0.371·21-s − 0.0167·23-s − 0.672·25-s − 1.01·27-s − 0.469·29-s − 0.417·31-s − 1.52·33-s + 0.216·35-s − 1.73·37-s − 0.360·39-s + 0.415·41-s + 1.85·43-s + 0.0185·45-s − 0.267·47-s + 1/7·49-s + 0.804·51-s − 0.724·53-s + 0.886·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 - 46 T + p^{7} T^{2} \) |
| 5 | \( 1 + 32 p T + p^{7} T^{2} \) |
| 11 | \( 1 + 6840 T + p^{7} T^{2} \) |
| 13 | \( 1 + 2900 T + p^{7} T^{2} \) |
| 17 | \( 1 - 16566 T + p^{7} T^{2} \) |
| 19 | \( 1 + 6718 T + p^{7} T^{2} \) |
| 23 | \( 1 + 976 T + p^{7} T^{2} \) |
| 29 | \( 1 + 61662 T + p^{7} T^{2} \) |
| 31 | \( 1 + 69236 T + p^{7} T^{2} \) |
| 37 | \( 1 + 533062 T + p^{7} T^{2} \) |
| 41 | \( 1 - 183158 T + p^{7} T^{2} \) |
| 43 | \( 1 - 966864 T + p^{7} T^{2} \) |
| 47 | \( 1 + 190268 T + p^{7} T^{2} \) |
| 53 | \( 1 + 785010 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2893594 T + p^{7} T^{2} \) |
| 61 | \( 1 + 95896 T + p^{7} T^{2} \) |
| 67 | \( 1 + 991644 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1068160 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2523458 T + p^{7} T^{2} \) |
| 79 | \( 1 - 285848 T + p^{7} T^{2} \) |
| 83 | \( 1 - 7094938 T + p^{7} T^{2} \) |
| 89 | \( 1 + 252390 T + p^{7} T^{2} \) |
| 97 | \( 1 + 1824794 T + p^{7} T^{2} \) |
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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
−13.30575396014611752441719127321, −12.25293640987987850059396425593, −10.77436420221191619201645279816, −9.541387429133680777365661702220, −8.235373873678361209102264697036, −7.44918648935519779576947778093, −5.46492166963714103440327209384, −3.63447058511631735748971482447, −2.43442260260108599569188206646, 0,
2.43442260260108599569188206646, 3.63447058511631735748971482447, 5.46492166963714103440327209384, 7.44918648935519779576947778093, 8.235373873678361209102264697036, 9.541387429133680777365661702220, 10.77436420221191619201645279816, 12.25293640987987850059396425593, 13.30575396014611752441719127321
