| L(s) = 1 | + 6·3-s − 4·5-s − 207·9-s − 240·11-s + 744·13-s − 24·15-s + 1.04e3·17-s + 986·19-s + 184·23-s − 3.10e3·25-s − 2.70e3·27-s − 734·29-s − 5.14e3·31-s − 1.44e3·33-s − 6.05e3·37-s + 4.46e3·39-s − 7.59e3·41-s + 1.30e4·43-s + 828·45-s − 1.46e4·47-s + 6.25e3·51-s − 1.45e4·53-s + 960·55-s + 5.91e3·57-s + 1.33e4·59-s − 9.67e3·61-s − 2.97e3·65-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 0.0715·5-s − 0.851·9-s − 0.598·11-s + 1.22·13-s − 0.0275·15-s + 0.874·17-s + 0.626·19-s + 0.0725·23-s − 0.994·25-s − 0.712·27-s − 0.162·29-s − 0.960·31-s − 0.230·33-s − 0.727·37-s + 0.469·39-s − 0.705·41-s + 1.07·43-s + 0.0609·45-s − 0.968·47-s + 0.336·51-s − 0.710·53-s + 0.0427·55-s + 0.241·57-s + 0.499·59-s − 0.332·61-s − 0.0873·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 p T + p^{5} T^{2} \) |
| 5 | \( 1 + 4 T + p^{5} T^{2} \) |
| 11 | \( 1 + 240 T + p^{5} T^{2} \) |
| 13 | \( 1 - 744 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1042 T + p^{5} T^{2} \) |
| 19 | \( 1 - 986 T + p^{5} T^{2} \) |
| 23 | \( 1 - 8 p T + p^{5} T^{2} \) |
| 29 | \( 1 + 734 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5140 T + p^{5} T^{2} \) |
| 37 | \( 1 + 6054 T + p^{5} T^{2} \) |
| 41 | \( 1 + 7598 T + p^{5} T^{2} \) |
| 43 | \( 1 - 13016 T + p^{5} T^{2} \) |
| 47 | \( 1 + 14668 T + p^{5} T^{2} \) |
| 53 | \( 1 + 274 p T + p^{5} T^{2} \) |
| 59 | \( 1 - 13362 T + p^{5} T^{2} \) |
| 61 | \( 1 + 9676 T + p^{5} T^{2} \) |
| 67 | \( 1 + 62124 T + p^{5} T^{2} \) |
| 71 | \( 1 + 2112 T + p^{5} T^{2} \) |
| 73 | \( 1 - 28910 T + p^{5} T^{2} \) |
| 79 | \( 1 + 101768 T + p^{5} T^{2} \) |
| 83 | \( 1 - 23922 T + p^{5} T^{2} \) |
| 89 | \( 1 + 141674 T + p^{5} T^{2} \) |
| 97 | \( 1 + 99982 T + p^{5} T^{2} \) |
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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
−9.999099280463680372452111832969, −9.011821428976211718539830646473, −8.206011537136489736738557165445, −7.42571245193174346406757090594, −6.03341100595289691149304756517, −5.32562654044314617085389727467, −3.78086331282893959733255985966, −2.95252294424584497146402127907, −1.53437697583797549596915824703, 0,
1.53437697583797549596915824703, 2.95252294424584497146402127907, 3.78086331282893959733255985966, 5.32562654044314617085389727467, 6.03341100595289691149304756517, 7.42571245193174346406757090594, 8.206011537136489736738557165445, 9.011821428976211718539830646473, 9.999099280463680372452111832969
