| L(s) = 1 | − 3·7-s − 8-s − 3·11-s + 12·19-s + 6·23-s − 12·29-s − 6·31-s − 6·41-s + 6·43-s + 24·47-s + 6·49-s + 3·56-s + 24·59-s + 18·61-s − 7·64-s + 12·67-s − 12·71-s + 24·73-s + 9·77-s + 12·79-s + 18·83-s + 3·88-s − 18·89-s + 24·97-s + 12·101-s + 12·107-s − 12·109-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 0.353·8-s − 0.904·11-s + 2.75·19-s + 1.25·23-s − 2.22·29-s − 1.07·31-s − 0.937·41-s + 0.914·43-s + 3.50·47-s + 6/7·49-s + 0.400·56-s + 3.12·59-s + 2.30·61-s − 7/8·64-s + 1.46·67-s − 1.42·71-s + 2.80·73-s + 1.02·77-s + 1.35·79-s + 1.97·83-s + 0.319·88-s − 1.90·89-s + 2.43·97-s + 1.19·101-s + 1.16·107-s − 1.14·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.066356348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066356348\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 + T^{3} + p^{3} T^{6} \) |
| 5 | $D_{6}$ | \( 1 - 2 T^{3} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 24 T^{2} + 2 T^{3} + 24 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 27 T^{2} - 8 T^{3} + 27 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 12 T + 84 T^{2} - 420 T^{3} + 84 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 57 T^{2} - 244 T^{3} + 57 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 12 T + 120 T^{2} + 702 T^{3} + 120 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 57 T^{2} + 404 T^{3} + 57 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 36 T^{2} - 246 T^{3} + 36 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 51 T^{2} + 460 T^{3} + 51 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 6 T + 117 T^{2} - 468 T^{3} + 117 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 24 T + 312 T^{2} - 2584 T^{3} + 312 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 111 T^{2} + 120 T^{3} + 111 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 24 T + 264 T^{2} - 2116 T^{3} + 264 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 228 T^{2} - 1604 T^{3} + 228 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 165 T^{2} + 1320 T^{3} + 165 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 24 T + 396 T^{2} - 3898 T^{3} + 396 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 189 T^{2} - 1640 T^{3} + 189 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 18 T + 309 T^{2} - 3036 T^{3} + 309 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 18 T + 183 T^{2} + 1308 T^{3} + 183 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 24 T + 435 T^{2} - 4664 T^{3} + 435 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.350735673943346928605407243924, −9.009891951703012985039789877855, −8.791199579904368345814017220001, −8.751931211758680633962335380238, −8.016988979946331964722263858808, −7.79978326617433976009661469734, −7.29271467714987259448403057269, −7.27945009656910062558633431494, −7.26462857320688202185871362338, −6.61026356290737500466431849547, −6.42103710521066609766729256108, −5.80565782620660755873390959829, −5.57451343138068983183318816028, −5.30898096073083919843041334668, −5.30200701719432773777941487258, −4.84980053666902816021388289595, −4.06983760073111197019964568070, −3.76758633823650279172768131407, −3.50835113085756827866004396710, −3.32704169591219710646693010646, −2.71824404546642850855812380802, −2.29477518003028835573674020964, −2.07452474876015681857580409472, −0.849121879007399067292849754039, −0.74665830070033025775674089900,
0.74665830070033025775674089900, 0.849121879007399067292849754039, 2.07452474876015681857580409472, 2.29477518003028835573674020964, 2.71824404546642850855812380802, 3.32704169591219710646693010646, 3.50835113085756827866004396710, 3.76758633823650279172768131407, 4.06983760073111197019964568070, 4.84980053666902816021388289595, 5.30200701719432773777941487258, 5.30898096073083919843041334668, 5.57451343138068983183318816028, 5.80565782620660755873390959829, 6.42103710521066609766729256108, 6.61026356290737500466431849547, 7.26462857320688202185871362338, 7.27945009656910062558633431494, 7.29271467714987259448403057269, 7.79978326617433976009661469734, 8.016988979946331964722263858808, 8.751931211758680633962335380238, 8.791199579904368345814017220001, 9.009891951703012985039789877855, 9.350735673943346928605407243924
