| L(s) = 1 | + 2-s + 9·3-s − 31·4-s + 34·5-s + 9·6-s − 63·8-s + 81·9-s + 34·10-s − 340·11-s − 279·12-s − 454·13-s + 306·15-s + 929·16-s + 798·17-s + 81·18-s − 892·19-s − 1.05e3·20-s − 340·22-s − 3.19e3·23-s − 567·24-s − 1.96e3·25-s − 454·26-s + 729·27-s − 8.24e3·29-s + 306·30-s + 2.49e3·31-s + 2.94e3·32-s + ⋯ |
| L(s) = 1 | + 0.176·2-s + 0.577·3-s − 0.968·4-s + 0.608·5-s + 0.102·6-s − 0.348·8-s + 1/3·9-s + 0.107·10-s − 0.847·11-s − 0.559·12-s − 0.745·13-s + 0.351·15-s + 0.907·16-s + 0.669·17-s + 0.0589·18-s − 0.566·19-s − 0.589·20-s − 0.149·22-s − 1.25·23-s − 0.200·24-s − 0.630·25-s − 0.131·26-s + 0.192·27-s − 1.81·29-s + 0.0620·30-s + 0.466·31-s + 0.508·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p^{5} T^{2} \) |
| 5 | \( 1 - 34 T + p^{5} T^{2} \) |
| 11 | \( 1 + 340 T + p^{5} T^{2} \) |
| 13 | \( 1 + 454 T + p^{5} T^{2} \) |
| 17 | \( 1 - 798 T + p^{5} T^{2} \) |
| 19 | \( 1 + 892 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3192 T + p^{5} T^{2} \) |
| 29 | \( 1 + 8242 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2496 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9798 T + p^{5} T^{2} \) |
| 41 | \( 1 + 19834 T + p^{5} T^{2} \) |
| 43 | \( 1 + 17236 T + p^{5} T^{2} \) |
| 47 | \( 1 + 8928 T + p^{5} T^{2} \) |
| 53 | \( 1 - 150 T + p^{5} T^{2} \) |
| 59 | \( 1 - 42396 T + p^{5} T^{2} \) |
| 61 | \( 1 + 14758 T + p^{5} T^{2} \) |
| 67 | \( 1 + 1676 T + p^{5} T^{2} \) |
| 71 | \( 1 - 14568 T + p^{5} T^{2} \) |
| 73 | \( 1 + 78378 T + p^{5} T^{2} \) |
| 79 | \( 1 + 2272 T + p^{5} T^{2} \) |
| 83 | \( 1 - 37764 T + p^{5} T^{2} \) |
| 89 | \( 1 - 117286 T + p^{5} T^{2} \) |
| 97 | \( 1 + 10002 T + p^{5} T^{2} \) |
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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
−11.86513712832689755367081880694, −10.11659301878757664520207170774, −9.747780081856411856801674783415, −8.492907024772248536017142691091, −7.62811003716241423533081602782, −5.91994262177281175199039814373, −4.86833271810734365365462657166, −3.53461218191255468799712939644, −2.02893298333085837447816606725, 0,
2.02893298333085837447816606725, 3.53461218191255468799712939644, 4.86833271810734365365462657166, 5.91994262177281175199039814373, 7.62811003716241423533081602782, 8.492907024772248536017142691091, 9.747780081856411856801674783415, 10.11659301878757664520207170774, 11.86513712832689755367081880694
