| L(s) = 1 | − 2·2-s + 3·3-s + 2·4-s − 6·6-s − 4·7-s − 8-s + 6·9-s − 8·11-s + 6·12-s − 4·13-s + 8·14-s − 4·16-s − 4·17-s − 12·18-s − 2·19-s − 12·21-s + 16·22-s − 6·23-s − 3·24-s + 8·26-s + 10·27-s − 8·28-s + 3·29-s + 6·31-s + 8·32-s − 24·33-s + 8·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.73·3-s + 4-s − 2.44·6-s − 1.51·7-s − 0.353·8-s + 2·9-s − 2.41·11-s + 1.73·12-s − 1.10·13-s + 2.13·14-s − 16-s − 0.970·17-s − 2.82·18-s − 0.458·19-s − 2.61·21-s + 3.41·22-s − 1.25·23-s − 0.612·24-s + 1.56·26-s + 1.92·27-s − 1.51·28-s + 0.557·29-s + 1.07·31-s + 1.41·32-s − 4.17·33-s + 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + p T + p T^{2} + T^{3} + p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 20 T^{2} + 48 T^{3} + 20 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 8 T + 48 T^{2} + 180 T^{3} + 48 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 + 4 T + 24 T^{2} + 42 T^{3} + 24 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 37 T^{2} + 92 T^{3} + 37 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 65 T^{2} + 244 T^{3} + 65 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 89 T^{2} - 340 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 584 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 23 T^{2} + 220 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 33 T^{2} - 88 T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 12 T + 132 T^{2} - 912 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 8 T + 55 T^{2} + 600 T^{3} + 55 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 20 T + 285 T^{2} + 2472 T^{3} + 285 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 4 T + 167 T^{2} - 432 T^{3} + 167 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 144 T^{2} - 52 T^{3} + 144 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 14 T + 153 T^{2} + 1572 T^{3} + 153 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 1160 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 177 T^{2} + 92 T^{3} + 177 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 221 T^{2} - 1120 T^{3} + 221 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 8 T + 136 T^{2} + 1350 T^{3} + 136 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 4 T + 219 T^{2} + 880 T^{3} + 219 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.431305038387335236144784036919, −8.101291546595078595811476868119, −8.020941212731803498981576918989, −7.69035773783767094304743006608, −7.67488377894685106830099142240, −7.38593470050147278941715019357, −6.82495239545744299998781286210, −6.71621709138969221965256576550, −6.64539560275474378142995615595, −6.28684810056489365318518376847, −5.92136522411034947831026664141, −5.44192275562190604860964137166, −5.23722153596370632844623904658, −4.84166293738482365120223238417, −4.55414215383279303450642025398, −4.33657393193564666165096057388, −3.94241505719236592565263668498, −3.55969777565999339269089988445, −3.24644437790843736065218801401, −2.76597027495677664721069095597, −2.49006891216790608009569272847, −2.47920582822239129684818280402, −2.38513349963122986926200377200, −1.49509356174790438486979206418, −1.39325491799662186369431614304, 0, 0, 0,
1.39325491799662186369431614304, 1.49509356174790438486979206418, 2.38513349963122986926200377200, 2.47920582822239129684818280402, 2.49006891216790608009569272847, 2.76597027495677664721069095597, 3.24644437790843736065218801401, 3.55969777565999339269089988445, 3.94241505719236592565263668498, 4.33657393193564666165096057388, 4.55414215383279303450642025398, 4.84166293738482365120223238417, 5.23722153596370632844623904658, 5.44192275562190604860964137166, 5.92136522411034947831026664141, 6.28684810056489365318518376847, 6.64539560275474378142995615595, 6.71621709138969221965256576550, 6.82495239545744299998781286210, 7.38593470050147278941715019357, 7.67488377894685106830099142240, 7.69035773783767094304743006608, 8.020941212731803498981576918989, 8.101291546595078595811476868119, 8.431305038387335236144784036919
