| L(s) = 1 | − 6·3-s + 4·5-s + 49·7-s − 207·9-s − 240·11-s − 744·13-s − 24·15-s − 1.04e3·17-s − 986·19-s − 294·21-s + 184·23-s − 3.10e3·25-s + 2.70e3·27-s − 734·29-s + 5.14e3·31-s + 1.44e3·33-s + 196·35-s − 6.05e3·37-s + 4.46e3·39-s + 7.59e3·41-s + 1.30e4·43-s − 828·45-s + 1.46e4·47-s + 2.40e3·49-s + 6.25e3·51-s − 1.45e4·53-s − 960·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 0.0715·5-s + 0.377·7-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 − 0.145·21-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 + 0.0270·35-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 + 1/7·49-s + 0.336·51-s − 0.710·53-s − 0.0427·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 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
−13.74939302447349727319812188133, −12.43492069778585565529671615478, −11.39725342837704550311809898810, −10.32156866516467691657262144944, −8.863654954865900799569635317304, −7.54620721974366629144754269528, −5.96908091612444019369391935758, −4.66208333584402984742175329908, −2.46125038256428076025062695303, 0,
2.46125038256428076025062695303, 4.66208333584402984742175329908, 5.96908091612444019369391935758, 7.54620721974366629144754269528, 8.863654954865900799569635317304, 10.32156866516467691657262144944, 11.39725342837704550311809898810, 12.43492069778585565529671615478, 13.74939302447349727319812188133
