| L(s) = 1 | + 9·3-s − 54·5-s − 49·7-s + 81·9-s − 216·11-s + 998·13-s − 486·15-s + 1.30e3·17-s − 884·19-s − 441·21-s + 2.26e3·23-s − 209·25-s + 729·27-s − 1.48e3·29-s − 8.36e3·31-s − 1.94e3·33-s + 2.64e3·35-s − 4.71e3·37-s + 8.98e3·39-s − 9.78e3·41-s − 1.94e4·43-s − 4.37e3·45-s − 2.22e4·47-s + 2.40e3·49-s + 1.17e4·51-s + 2.67e4·53-s + 1.16e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.965·5-s − 0.377·7-s + 1/3·9-s − 0.538·11-s + 1.63·13-s − 0.557·15-s + 1.09·17-s − 0.561·19-s − 0.218·21-s + 0.893·23-s − 0.0668·25-s + 0.192·27-s − 0.327·29-s − 1.56·31-s − 0.310·33-s + 0.365·35-s − 0.566·37-s + 0.945·39-s − 0.909·41-s − 1.60·43-s − 0.321·45-s − 1.46·47-s + 1/7·49-s + 0.630·51-s + 1.31·53-s + 0.519·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 54 T + p^{5} T^{2} \) |
| 11 | \( 1 + 216 T + p^{5} T^{2} \) |
| 13 | \( 1 - 998 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1302 T + p^{5} T^{2} \) |
| 19 | \( 1 + 884 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2268 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1482 T + p^{5} T^{2} \) |
| 31 | \( 1 + 8360 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4714 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9786 T + p^{5} T^{2} \) |
| 43 | \( 1 + 452 p T + p^{5} T^{2} \) |
| 47 | \( 1 + 22200 T + p^{5} T^{2} \) |
| 53 | \( 1 - 26790 T + p^{5} T^{2} \) |
| 59 | \( 1 + 28092 T + p^{5} T^{2} \) |
| 61 | \( 1 + 38866 T + p^{5} T^{2} \) |
| 67 | \( 1 + 23948 T + p^{5} T^{2} \) |
| 71 | \( 1 - 20628 T + p^{5} T^{2} \) |
| 73 | \( 1 - 290 T + p^{5} T^{2} \) |
| 79 | \( 1 - 99544 T + p^{5} T^{2} \) |
| 83 | \( 1 + 19308 T + p^{5} T^{2} \) |
| 89 | \( 1 - 36390 T + p^{5} T^{2} \) |
| 97 | \( 1 + 79078 T + p^{5} T^{2} \) |
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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
−10.34525349411340474063613337802, −9.141602499999592527916877999320, −8.326069478953102735885637577982, −7.59138029131357438011350894077, −6.50141122720935994797397279887, −5.22271954224789082020849324452, −3.77135720912009225667630729517, −3.25352601493356437605338940429, −1.52418914081202387421157297953, 0,
1.52418914081202387421157297953, 3.25352601493356437605338940429, 3.77135720912009225667630729517, 5.22271954224789082020849324452, 6.50141122720935994797397279887, 7.59138029131357438011350894077, 8.326069478953102735885637577982, 9.141602499999592527916877999320, 10.34525349411340474063613337802
