| L(s) = 1 | + 3·5-s + 3·7-s + 6·11-s − 3·13-s + 3·19-s + 6·23-s + 15·29-s − 3·31-s + 9·35-s − 3·37-s + 6·41-s + 3·43-s + 15·47-s + 6·49-s + 18·53-s + 18·55-s + 3·59-s − 6·61-s − 9·65-s − 6·67-s + 15·71-s + 9·73-s + 18·77-s + 3·79-s + 18·83-s − 6·89-s − 9·91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.13·7-s + 1.80·11-s − 0.832·13-s + 0.688·19-s + 1.25·23-s + 2.78·29-s − 0.538·31-s + 1.52·35-s − 0.493·37-s + 0.937·41-s + 0.457·43-s + 2.18·47-s + 6/7·49-s + 2.47·53-s + 2.42·55-s + 0.390·59-s − 0.768·61-s − 1.11·65-s − 0.733·67-s + 1.78·71-s + 1.05·73-s + 2.05·77-s + 0.337·79-s + 1.97·83-s − 0.635·89-s − 0.943·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.493766580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.493766580\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 3 T + 9 T^{2} - 21 T^{3} + 9 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 36 T^{2} - 123 T^{3} + 36 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 + 18 T^{2} - 9 T^{3} + 18 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 21 T^{2} - 65 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 54 T^{2} - 177 T^{3} + 54 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 15 T + 135 T^{2} - 807 T^{3} + 135 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 133 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 115 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 126 T^{2} - 483 T^{3} + 126 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 3 T + 21 T^{2} + 205 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 15 T + 177 T^{2} - 1329 T^{3} + 177 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 18 T + 198 T^{2} - 1521 T^{3} + 198 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 3 T + 123 T^{2} - 435 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 138 T^{2} + 763 T^{3} + 138 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 174 T^{2} + 745 T^{3} + 174 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 15 T + 231 T^{2} - 1833 T^{3} + 231 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 207 T^{2} - 1235 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 3 T + 3 T^{2} + 889 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 18 T + 324 T^{2} - 3015 T^{3} + 324 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 72 T^{2} - 21 T^{3} + 72 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 15 T + 309 T^{2} + 2887 T^{3} + 309 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.137358786635108106077005880484, −7.55990863272468300779993836038, −7.51318895597235508564355095841, −7.27750147617083737871357181065, −6.73876414438631983050789516627, −6.73661088880707328437725493218, −6.64384023419566450006948882387, −6.03826212504594097381309755226, −5.86810674829300134222377277796, −5.62765808299818400022755349188, −5.35072332053585461880387847954, −5.10465701277197981462820352526, −4.78547937322590141436847249787, −4.43929209960523638907577042458, −4.19532063894019818903498060781, −4.08227221565028679776404096391, −3.41102251552100291009783787170, −3.27460322272164033596761099256, −2.84344383354613276560775416923, −2.25134314512312011179343778810, −2.13133725884163376078313399610, −2.07474786433568748434270171766, −1.12904004686006969109738342785, −1.04483332189833138687378324111, −0.840729489854907882346668598271,
0.840729489854907882346668598271, 1.04483332189833138687378324111, 1.12904004686006969109738342785, 2.07474786433568748434270171766, 2.13133725884163376078313399610, 2.25134314512312011179343778810, 2.84344383354613276560775416923, 3.27460322272164033596761099256, 3.41102251552100291009783787170, 4.08227221565028679776404096391, 4.19532063894019818903498060781, 4.43929209960523638907577042458, 4.78547937322590141436847249787, 5.10465701277197981462820352526, 5.35072332053585461880387847954, 5.62765808299818400022755349188, 5.86810674829300134222377277796, 6.03826212504594097381309755226, 6.64384023419566450006948882387, 6.73661088880707328437725493218, 6.73876414438631983050789516627, 7.27750147617083737871357181065, 7.51318895597235508564355095841, 7.55990863272468300779993836038, 8.137358786635108106077005880484
