| L(s) = 1 | + 3·3-s − 2·4-s + 2·5-s + 8-s + 6·9-s − 2·11-s − 6·12-s − 4·13-s + 6·15-s − 3·17-s − 7·19-s − 4·20-s + 23-s + 3·24-s − 6·25-s + 10·27-s − 11·29-s − 13·31-s − 4·32-s − 6·33-s − 12·36-s − 5·37-s − 12·39-s + 2·40-s + 41-s + 6·43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 4-s + 0.894·5-s + 0.353·8-s + 2·9-s − 0.603·11-s − 1.73·12-s − 1.10·13-s + 1.54·15-s − 0.727·17-s − 1.60·19-s − 0.894·20-s + 0.208·23-s + 0.612·24-s − 6/5·25-s + 1.92·27-s − 2.04·29-s − 2.33·31-s − 0.707·32-s − 1.04·33-s − 2·36-s − 0.821·37-s − 1.92·39-s + 0.316·40-s + 0.156·41-s + 0.914·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | | \( 1 \) |
| 59 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p T^{2} - T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 2 T + 2 p T^{2} - 18 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 2 p T^{2} + 48 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 32 T^{2} + 6 p T^{3} + 32 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 8 T^{2} + 4 T^{3} + 8 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 7 T + 68 T^{2} + 270 T^{3} + 68 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - T + 42 T^{2} + 18 T^{3} + 42 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 11 T + 96 T^{2} + 564 T^{3} + 96 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 13 T + 130 T^{2} + 778 T^{3} + 130 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 5 T + 92 T^{2} + 384 T^{3} + 92 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - T + 84 T^{2} - 156 T^{3} + 84 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 6 T + 38 T^{2} + 76 T^{3} + 38 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 11 T + 104 T^{2} + 538 T^{3} + 104 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 2 T + 70 T^{2} - 270 T^{3} + 70 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - T + 82 T^{2} - 220 T^{3} + 82 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 10 T + 82 T^{2} - 556 T^{3} + 82 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 26 T + 406 T^{2} - 4116 T^{3} + 406 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 7 T + 78 T^{2} + 304 T^{3} + 78 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 206 T^{2} - 348 T^{3} + 206 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 3 T + 50 T^{2} - 350 T^{3} + 50 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 23 T + 358 T^{2} - 3816 T^{3} + 358 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 14 T + 266 T^{2} + 2514 T^{3} + 266 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.34569766308709797703524943810, −7.10894563970459596204884485945, −6.85895449921188364729545289046, −6.45717773568206876240028557709, −6.41211865677178124708375814013, −5.95075096880284746665393305254, −5.94059412692920673082984645163, −5.35580993370866890714227470055, −5.25106565334577342381765696357, −5.21010578659312736410895832452, −4.73343046487369062676550435771, −4.64881399579654932594346751538, −4.41413006860113493380529602206, −3.89988935142001638828638390247, −3.77625172925309890493712244417, −3.75447872322173883508394121281, −3.35258131603073782854895665998, −3.28316035111524549383533388159, −2.45569670005543527882777179231, −2.36551877750654504094480722234, −2.21064219120284570139066706989, −2.13786473328739330827915665288, −1.87119605078485821185271418488, −1.34889737279906887126548120343, −1.10256577815382894953400238593, 0, 0, 0,
1.10256577815382894953400238593, 1.34889737279906887126548120343, 1.87119605078485821185271418488, 2.13786473328739330827915665288, 2.21064219120284570139066706989, 2.36551877750654504094480722234, 2.45569670005543527882777179231, 3.28316035111524549383533388159, 3.35258131603073782854895665998, 3.75447872322173883508394121281, 3.77625172925309890493712244417, 3.89988935142001638828638390247, 4.41413006860113493380529602206, 4.64881399579654932594346751538, 4.73343046487369062676550435771, 5.21010578659312736410895832452, 5.25106565334577342381765696357, 5.35580993370866890714227470055, 5.94059412692920673082984645163, 5.95075096880284746665393305254, 6.41211865677178124708375814013, 6.45717773568206876240028557709, 6.85895449921188364729545289046, 7.10894563970459596204884485945, 7.34569766308709797703524943810
