| L(s) = 1 | − 2·3-s + 4·5-s + 9-s + 4·11-s − 8·15-s − 4·17-s − 8·19-s + 4·23-s + 12·25-s + 2·27-s − 16·31-s − 8·33-s + 8·37-s + 8·41-s + 32·43-s + 4·45-s − 8·47-s + 8·51-s + 4·53-s + 16·55-s + 16·57-s − 16·59-s + 8·61-s − 16·67-s − 8·69-s − 8·71-s − 8·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 1/3·9-s + 1.20·11-s − 2.06·15-s − 0.970·17-s − 1.83·19-s + 0.834·23-s + 12/5·25-s + 0.384·27-s − 2.87·31-s − 1.39·33-s + 1.31·37-s + 1.24·41-s + 4.87·43-s + 0.596·45-s − 1.16·47-s + 1.12·51-s + 0.549·53-s + 2.15·55-s + 2.11·57-s − 2.08·59-s + 1.02·61-s − 1.95·67-s − 0.963·69-s − 0.949·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.063409690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.063409690\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T - 2 T^{2} + 16 T^{3} + 27 T^{4} + 16 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T - 4 T^{2} - 56 T^{3} - 161 T^{4} - 56 p T^{5} - 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 8 T + 18 T^{2} + 64 T^{3} + 539 T^{4} + 64 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 - 4 T - 26 T^{2} + 16 T^{3} + 867 T^{4} + 16 p T^{5} - 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 16 T + 138 T^{2} + 896 T^{3} + 5027 T^{4} + 896 p T^{5} + 138 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 8 T + 6 T^{2} + 128 T^{3} - 373 T^{4} + 128 p T^{5} + 6 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 84 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8 T - 38 T^{2} + 64 T^{3} + 5187 T^{4} + 64 p T^{5} - 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T + 34 T^{2} + 496 T^{3} - 3333 T^{4} + 496 p T^{5} + 34 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 16 T + 82 T^{2} + 896 T^{3} + 11691 T^{4} + 896 p T^{5} + 82 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T + 24 T^{2} + 656 T^{3} - 6025 T^{4} + 656 p T^{5} + 24 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 8 T - 48 T^{2} - 272 T^{3} + 2543 T^{4} - 272 p T^{5} - 48 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 512 p T^{5} + 66 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 4 T - 4 T^{2} + 632 T^{3} - 8945 T^{4} + 632 p T^{5} - 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.18213052107709928568859641325, −6.12281696972679269027015061820, −6.03803854465560626795340341940, −5.83653026533453844007947924983, −5.71046951547934159512374800317, −5.53071345208376988329340464057, −5.05226268992738086202951479910, −5.00424547992461019113852662489, −4.72415859287628441558232217898, −4.42843307425872158627013801696, −4.32401482583756490396672930322, −4.15298965405557485398773736912, −4.06784138651864694687367080439, −3.52451004385857063902016839595, −3.23815941043690269312109284592, −3.09826541249015753483816918029, −2.78405286191202133044887024297, −2.37710621827355710012074247639, −2.29171940968757990587790912383, −1.95374667719756834656581477402, −1.73503280561763342349569219973, −1.58544750883032014644645527774, −0.889822657573869278369897261474, −0.808663742352786696135307766470, −0.41350071560858887148526125659,
0.41350071560858887148526125659, 0.808663742352786696135307766470, 0.889822657573869278369897261474, 1.58544750883032014644645527774, 1.73503280561763342349569219973, 1.95374667719756834656581477402, 2.29171940968757990587790912383, 2.37710621827355710012074247639, 2.78405286191202133044887024297, 3.09826541249015753483816918029, 3.23815941043690269312109284592, 3.52451004385857063902016839595, 4.06784138651864694687367080439, 4.15298965405557485398773736912, 4.32401482583756490396672930322, 4.42843307425872158627013801696, 4.72415859287628441558232217898, 5.00424547992461019113852662489, 5.05226268992738086202951479910, 5.53071345208376988329340464057, 5.71046951547934159512374800317, 5.83653026533453844007947924983, 6.03803854465560626795340341940, 6.12281696972679269027015061820, 6.18213052107709928568859641325
