| L(s) = 1 | − 27·3-s − 378·5-s − 832·7-s + 729·9-s − 2.48e3·11-s + 1.48e4·13-s + 1.02e4·15-s − 2.23e4·17-s − 1.63e4·19-s + 2.24e4·21-s − 1.15e5·23-s + 6.47e4·25-s − 1.96e4·27-s + 1.57e5·29-s − 1.64e4·31-s + 6.70e4·33-s + 3.14e5·35-s − 1.49e5·37-s − 4.01e5·39-s − 2.41e5·41-s − 4.43e5·43-s − 2.75e5·45-s + 9.22e5·47-s − 1.31e5·49-s + 6.02e5·51-s − 6.97e5·53-s + 9.38e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.35·5-s − 0.916·7-s + 1/3·9-s − 0.562·11-s + 1.87·13-s + 0.780·15-s − 1.10·17-s − 0.545·19-s + 0.529·21-s − 1.97·23-s + 0.828·25-s − 0.192·27-s + 1.19·29-s − 0.0992·31-s + 0.324·33-s + 1.23·35-s − 0.484·37-s − 1.08·39-s − 0.546·41-s − 0.850·43-s − 0.450·45-s + 1.29·47-s − 0.159·49-s + 0.635·51-s − 0.643·53-s + 0.760·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| good | 5 | \( 1 + 378 T + p^{7} T^{2} \) |
| 7 | \( 1 + 832 T + p^{7} T^{2} \) |
| 11 | \( 1 + 2484 T + p^{7} T^{2} \) |
| 13 | \( 1 - 14870 T + p^{7} T^{2} \) |
| 17 | \( 1 + 22302 T + p^{7} T^{2} \) |
| 19 | \( 1 + 16300 T + p^{7} T^{2} \) |
| 23 | \( 1 + 115128 T + p^{7} T^{2} \) |
| 29 | \( 1 - 157086 T + p^{7} T^{2} \) |
| 31 | \( 1 + 16456 T + p^{7} T^{2} \) |
| 37 | \( 1 + 149266 T + p^{7} T^{2} \) |
| 41 | \( 1 + 241110 T + p^{7} T^{2} \) |
| 43 | \( 1 + 443188 T + p^{7} T^{2} \) |
| 47 | \( 1 - 922752 T + p^{7} T^{2} \) |
| 53 | \( 1 + 697626 T + p^{7} T^{2} \) |
| 59 | \( 1 - 870156 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2067062 T + p^{7} T^{2} \) |
| 67 | \( 1 + 1680748 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1070280 T + p^{7} T^{2} \) |
| 73 | \( 1 + 2403334 T + p^{7} T^{2} \) |
| 79 | \( 1 - 2301512 T + p^{7} T^{2} \) |
| 83 | \( 1 - 4708692 T + p^{7} T^{2} \) |
| 89 | \( 1 - 4143690 T + p^{7} T^{2} \) |
| 97 | \( 1 + 1622974 T + p^{7} T^{2} \) |
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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
−18.03856653775611897044462968328, −16.04236693790318248664141314339, −15.72040965290009028364985978909, −13.33585711915779779560707592758, −11.91508904072488664816649049968, −10.60870359949416380395058553144, −8.347822757510313655311611468920, −6.41907305343413758947245046321, −3.92006500247607182459096140991, 0,
3.92006500247607182459096140991, 6.41907305343413758947245046321, 8.347822757510313655311611468920, 10.60870359949416380395058553144, 11.91508904072488664816649049968, 13.33585711915779779560707592758, 15.72040965290009028364985978909, 16.04236693790318248664141314339, 18.03856653775611897044462968328
