| L(s) = 1 | − 27·3-s − 390·5-s + 64·7-s + 729·9-s − 948·11-s + 5.09e3·13-s + 1.05e4·15-s + 2.83e4·17-s − 8.62e3·19-s − 1.72e3·21-s + 1.52e4·23-s + 7.39e4·25-s − 1.96e4·27-s − 3.65e4·29-s + 2.76e5·31-s + 2.55e4·33-s − 2.49e4·35-s − 2.68e5·37-s − 1.37e5·39-s − 6.29e5·41-s + 6.85e5·43-s − 2.84e5·45-s − 5.83e5·47-s − 8.19e5·49-s − 7.66e5·51-s + 4.28e5·53-s + 3.69e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.39·5-s + 0.0705·7-s + 1/3·9-s − 0.214·11-s + 0.643·13-s + 0.805·15-s + 1.40·17-s − 0.288·19-s − 0.0407·21-s + 0.262·23-s + 0.946·25-s − 0.192·27-s − 0.277·29-s + 1.66·31-s + 0.123·33-s − 0.0984·35-s − 0.871·37-s − 0.371·39-s − 1.42·41-s + 1.31·43-s − 0.465·45-s − 0.819·47-s − 0.995·49-s − 0.809·51-s + 0.394·53-s + 0.299·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 78 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 64 T + p^{7} T^{2} \) |
| 11 | \( 1 + 948 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5098 T + p^{7} T^{2} \) |
| 17 | \( 1 - 28386 T + p^{7} T^{2} \) |
| 19 | \( 1 + 8620 T + p^{7} T^{2} \) |
| 23 | \( 1 - 15288 T + p^{7} T^{2} \) |
| 29 | \( 1 + 36510 T + p^{7} T^{2} \) |
| 31 | \( 1 - 276808 T + p^{7} T^{2} \) |
| 37 | \( 1 + 268526 T + p^{7} T^{2} \) |
| 41 | \( 1 + 629718 T + p^{7} T^{2} \) |
| 43 | \( 1 - 685772 T + p^{7} T^{2} \) |
| 47 | \( 1 + 583296 T + p^{7} T^{2} \) |
| 53 | \( 1 - 428058 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1306380 T + p^{7} T^{2} \) |
| 61 | \( 1 + 300662 T + p^{7} T^{2} \) |
| 67 | \( 1 + 507244 T + p^{7} T^{2} \) |
| 71 | \( 1 + 5560632 T + p^{7} T^{2} \) |
| 73 | \( 1 - 1369082 T + p^{7} T^{2} \) |
| 79 | \( 1 - 6913720 T + p^{7} T^{2} \) |
| 83 | \( 1 + 4376748 T + p^{7} T^{2} \) |
| 89 | \( 1 + 8528310 T + p^{7} T^{2} \) |
| 97 | \( 1 + 8826814 T + p^{7} T^{2} \) |
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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
−10.92855235211862053863189827700, −9.974495480191317072657874708298, −8.481232644464964717831537732914, −7.74663214224141048740806794350, −6.66490151997038779333662745852, −5.37917673145249050354010592240, −4.22120609695061456796833116405, −3.19349468104231142538713506612, −1.18240424046115842984692125562, 0,
1.18240424046115842984692125562, 3.19349468104231142538713506612, 4.22120609695061456796833116405, 5.37917673145249050354010592240, 6.66490151997038779333662745852, 7.74663214224141048740806794350, 8.481232644464964717831537732914, 9.974495480191317072657874708298, 10.92855235211862053863189827700
