| L(s) = 1 | − 2·5-s + 2·7-s − 3·9-s − 6·13-s − 2·17-s − 8·23-s − 2·25-s − 6·29-s + 2·31-s − 4·35-s − 2·37-s + 6·41-s + 8·43-s + 6·45-s + 8·47-s + 6·49-s − 6·53-s − 16·59-s − 2·61-s − 6·63-s + 12·65-s − 4·73-s + 2·79-s + 9·81-s − 8·83-s + 4·85-s + 2·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 9-s − 1.66·13-s − 0.485·17-s − 1.66·23-s − 2/5·25-s − 1.11·29-s + 0.359·31-s − 0.676·35-s − 0.328·37-s + 0.937·41-s + 1.21·43-s + 0.894·45-s + 1.16·47-s + 6/7·49-s − 0.824·53-s − 2.08·59-s − 0.256·61-s − 0.755·63-s + 1.48·65-s − 0.468·73-s + 0.225·79-s + 81-s − 0.878·83-s + 0.433·85-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T - 2 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T - 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.2676851181, −14.8865362612, −14.2710054501, −14.0794545042, −13.7089242451, −12.8002977767, −12.3873085858, −11.9620847784, −11.6981479268, −11.0906962546, −10.7711751774, −10.1180388857, −9.44805195820, −9.10266798696, −8.39644576865, −7.85905025590, −7.58049245284, −7.12881767165, −6.01788896950, −5.79983462691, −4.89185910144, −4.38179104794, −3.77380982763, −2.72880096534, −2.04922142769, 0,
2.04922142769, 2.72880096534, 3.77380982763, 4.38179104794, 4.89185910144, 5.79983462691, 6.01788896950, 7.12881767165, 7.58049245284, 7.85905025590, 8.39644576865, 9.10266798696, 9.44805195820, 10.1180388857, 10.7711751774, 11.0906962546, 11.6981479268, 11.9620847784, 12.3873085858, 12.8002977767, 13.7089242451, 14.0794545042, 14.2710054501, 14.8865362612, 15.2676851181
