| L(s) = 1 | − 5-s − 5·11-s − 2·13-s + 4·17-s + 3·19-s − 14·23-s − 2·25-s + 5·29-s − 2·31-s − 12·41-s − 9·43-s + 9·47-s − 6·53-s + 5·55-s + 5·59-s + 7·61-s + 2·65-s − 16·67-s − 11·71-s − 73-s − 8·79-s − 17·83-s − 4·85-s + 3·89-s − 3·95-s + 14·97-s + 29·101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.50·11-s − 0.554·13-s + 0.970·17-s + 0.688·19-s − 2.91·23-s − 2/5·25-s + 0.928·29-s − 0.359·31-s − 1.87·41-s − 1.37·43-s + 1.31·47-s − 0.824·53-s + 0.674·55-s + 0.650·59-s + 0.896·61-s + 0.248·65-s − 1.95·67-s − 1.30·71-s − 0.117·73-s − 0.900·79-s − 1.86·83-s − 0.433·85-s + 0.317·89-s − 0.307·95-s + 1.42·97-s + 2.88·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{9} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{9} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $S_4\times C_2$ | \( 1 + T + 3 T^{2} + 19 T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 5 T + 21 T^{2} + 47 T^{3} + 21 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 28 T^{2} + 31 T^{3} + 28 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 36 T^{2} - 145 T^{3} + 36 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 + 14 T + 108 T^{2} + 581 T^{3} + 108 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 5 T + 75 T^{2} - 227 T^{3} + 75 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 82 T^{2} + 103 T^{3} + 82 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 54 T^{2} - 137 T^{3} + 54 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 12 T + 132 T^{2} + 903 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 9 T + 99 T^{2} + 493 T^{3} + 99 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 9 T + 87 T^{2} - 603 T^{3} + 87 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 69 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 5 T + 159 T^{2} - 509 T^{3} + 159 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 7 T + 94 T^{2} - 287 T^{3} + 94 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 16 T + 260 T^{2} + 2197 T^{3} + 260 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 11 T + 141 T^{2} + 1373 T^{3} + 141 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + T + 137 T^{2} + 307 T^{3} + 137 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 8 T + 82 T^{2} + 391 T^{3} + 82 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 17 T + 333 T^{2} + 2921 T^{3} + 333 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 3 T + 42 T^{2} + 789 T^{3} + 42 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 14 T + 251 T^{2} - 2660 T^{3} + 251 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.62745228813986881036240842941, −7.32516021429745462083159740903, −7.32419797786376529636340303125, −7.02512164304191333972797560873, −6.57410113715227257780263447972, −6.38302547442017877522158622667, −6.19493488319327953721802739545, −5.72810912513948336841552031241, −5.72327104311223751082166002486, −5.51707339873581468924384529957, −4.99399887334386100527941359003, −4.96238606197840698588569645860, −4.84920111685029984362614098339, −4.33974472449841863927900632222, −3.96955609806610687673505544811, −3.93504768854164924720256039757, −3.49149298479971412092632625351, −3.44138472961130754778480494350, −2.87075250750224091041316154275, −2.61290690353787990078751944375, −2.54413535985168282813361707824, −2.15377442946030206787119381951, −1.54954925435835195863304725043, −1.48065578480195045887138610000, −1.07492244614541847376880068682, 0, 0, 0,
1.07492244614541847376880068682, 1.48065578480195045887138610000, 1.54954925435835195863304725043, 2.15377442946030206787119381951, 2.54413535985168282813361707824, 2.61290690353787990078751944375, 2.87075250750224091041316154275, 3.44138472961130754778480494350, 3.49149298479971412092632625351, 3.93504768854164924720256039757, 3.96955609806610687673505544811, 4.33974472449841863927900632222, 4.84920111685029984362614098339, 4.96238606197840698588569645860, 4.99399887334386100527941359003, 5.51707339873581468924384529957, 5.72327104311223751082166002486, 5.72810912513948336841552031241, 6.19493488319327953721802739545, 6.38302547442017877522158622667, 6.57410113715227257780263447972, 7.02512164304191333972797560873, 7.32419797786376529636340303125, 7.32516021429745462083159740903, 7.62745228813986881036240842941
