| L(s) = 1 | − 9·2-s − 9·3-s + 49·4-s + 24·5-s + 81·6-s + 72·7-s − 153·8-s + 81·9-s − 216·10-s − 441·12-s + 306·13-s − 648·14-s − 216·15-s − 191·16-s + 1.20e3·17-s − 729·18-s − 774·19-s + 1.17e3·20-s − 648·21-s − 4.62e3·23-s + 1.37e3·24-s − 2.54e3·25-s − 2.75e3·26-s − 729·27-s + 3.52e3·28-s − 7.68e3·29-s + 1.94e3·30-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.577·3-s + 1.53·4-s + 0.429·5-s + 0.918·6-s + 0.555·7-s − 0.845·8-s + 1/3·9-s − 0.683·10-s − 0.884·12-s + 0.502·13-s − 0.883·14-s − 0.247·15-s − 0.186·16-s + 1.01·17-s − 0.530·18-s − 0.491·19-s + 0.657·20-s − 0.320·21-s − 1.82·23-s + 0.487·24-s − 0.815·25-s − 0.798·26-s − 0.192·27-s + 0.850·28-s − 1.69·29-s + 0.394·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{2} T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 9 T + p^{5} T^{2} \) |
| 5 | \( 1 - 24 T + p^{5} T^{2} \) |
| 7 | \( 1 - 72 T + p^{5} T^{2} \) |
| 13 | \( 1 - 306 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1206 T + p^{5} T^{2} \) |
| 19 | \( 1 + 774 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4626 T + p^{5} T^{2} \) |
| 29 | \( 1 + 7686 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5428 T + p^{5} T^{2} \) |
| 37 | \( 1 - 3454 T + p^{5} T^{2} \) |
| 41 | \( 1 - 7866 T + p^{5} T^{2} \) |
| 43 | \( 1 - 15786 T + p^{5} T^{2} \) |
| 47 | \( 1 + 6402 T + p^{5} T^{2} \) |
| 53 | \( 1 + 21684 T + p^{5} T^{2} \) |
| 59 | \( 1 + 27420 T + p^{5} T^{2} \) |
| 61 | \( 1 - 52866 T + p^{5} T^{2} \) |
| 67 | \( 1 - 25012 T + p^{5} T^{2} \) |
| 71 | \( 1 - 65058 T + p^{5} T^{2} \) |
| 73 | \( 1 + 26676 T + p^{5} T^{2} \) |
| 79 | \( 1 + 18612 T + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 + 41670 T + p^{5} T^{2} \) |
| 97 | \( 1 - 40694 T + p^{5} T^{2} \) |
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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
−9.946137289658365258593372063095, −9.457687897137528402867318932394, −8.157860603792140774304566615747, −7.72863711334436395479713052566, −6.42345153254255051011800912543, −5.60696348439850882561103453371, −4.08430797528903131527281778452, −2.15208152315674811919420244456, −1.24259946185683390241928148054, 0,
1.24259946185683390241928148054, 2.15208152315674811919420244456, 4.08430797528903131527281778452, 5.60696348439850882561103453371, 6.42345153254255051011800912543, 7.72863711334436395479713052566, 8.157860603792140774304566615747, 9.457687897137528402867318932394, 9.946137289658365258593372063095
