| L(s) = 1 | + 234·2-s + 2.19e4·4-s − 2.80e5·5-s − 1.37e6·7-s − 2.52e6·8-s − 6.56e7·10-s − 3.40e7·11-s + 3.84e8·13-s − 3.21e8·14-s − 1.31e9·16-s − 1.25e9·17-s − 2.49e9·19-s − 6.17e9·20-s − 7.96e9·22-s − 1.12e10·23-s + 4.82e10·25-s + 8.98e10·26-s − 3.01e10·28-s + 4.84e10·29-s + 1.30e11·31-s − 2.24e11·32-s − 2.94e11·34-s + 3.85e11·35-s − 2.00e11·37-s − 5.84e11·38-s + 7.08e11·40-s − 6.79e11·41-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.671·4-s − 1.60·5-s − 0.630·7-s − 0.425·8-s − 2.07·10-s − 0.526·11-s + 1.69·13-s − 0.814·14-s − 1.22·16-s − 0.744·17-s − 0.641·19-s − 1.07·20-s − 0.680·22-s − 0.691·23-s + 1.58·25-s + 2.19·26-s − 0.422·28-s + 0.521·29-s + 0.852·31-s − 1.15·32-s − 0.962·34-s + 1.01·35-s − 0.346·37-s − 0.829·38-s + 0.683·40-s − 0.544·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 117 p T + p^{15} T^{2} \) |
| 5 | \( 1 + 56142 p T + p^{15} T^{2} \) |
| 7 | \( 1 + 196192 p T + p^{15} T^{2} \) |
| 11 | \( 1 + 3093732 p T + p^{15} T^{2} \) |
| 13 | \( 1 - 29540174 p T + p^{15} T^{2} \) |
| 17 | \( 1 + 1259207586 T + p^{15} T^{2} \) |
| 19 | \( 1 + 2499071020 T + p^{15} T^{2} \) |
| 23 | \( 1 + 11284833672 T + p^{15} T^{2} \) |
| 29 | \( 1 - 48413458530 T + p^{15} T^{2} \) |
| 31 | \( 1 - 130547265752 T + p^{15} T^{2} \) |
| 37 | \( 1 + 200223317554 T + p^{15} T^{2} \) |
| 41 | \( 1 + 679141724202 T + p^{15} T^{2} \) |
| 43 | \( 1 - 279482194892 T + p^{15} T^{2} \) |
| 47 | \( 1 + 1520672832576 T + p^{15} T^{2} \) |
| 53 | \( 1 + 2646053822502 T + p^{15} T^{2} \) |
| 59 | \( 1 + 7399371294540 T + p^{15} T^{2} \) |
| 61 | \( 1 + 42659617819498 T + p^{15} T^{2} \) |
| 67 | \( 1 + 56408026065964 T + p^{15} T^{2} \) |
| 71 | \( 1 - 133149677299848 T + p^{15} T^{2} \) |
| 73 | \( 1 - 105603350884922 T + p^{15} T^{2} \) |
| 79 | \( 1 + 55665674361880 T + p^{15} T^{2} \) |
| 83 | \( 1 + 378077412997332 T + p^{15} T^{2} \) |
| 89 | \( 1 + 219315065897610 T + p^{15} T^{2} \) |
| 97 | \( 1 - 703322682162626 T + p^{15} T^{2} \) |
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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
−15.94975230373702367609684164490, −15.37957743871858346982311954044, −13.60228144530533174523323693562, −12.40150591253314084315642611579, −11.11667572095267042038072208799, −8.415536609027378731251635101217, −6.39821900210088567252431060057, −4.36320787250641448880570381774, −3.28524151911726830018044993257, 0,
3.28524151911726830018044993257, 4.36320787250641448880570381774, 6.39821900210088567252431060057, 8.415536609027378731251635101217, 11.11667572095267042038072208799, 12.40150591253314084315642611579, 13.60228144530533174523323693562, 15.37957743871858346982311954044, 15.94975230373702367609684164490
