| L(s) = 1 | − 6·3-s + 12·5-s + 10·7-s + 21·9-s + 12·11-s − 72·15-s − 48·17-s + 42·19-s − 60·21-s − 24·23-s + 58·25-s − 54·27-s − 102·31-s − 72·33-s + 120·35-s + 22·37-s − 28·43-s + 252·45-s + 132·47-s + 49·49-s + 288·51-s + 120·53-s + 144·55-s − 252·57-s + 24·59-s − 72·61-s + 210·63-s + ⋯ |
| L(s) = 1 | − 2·3-s + 12/5·5-s + 10/7·7-s + 7/3·9-s + 1.09·11-s − 4.79·15-s − 2.82·17-s + 2.21·19-s − 2.85·21-s − 1.04·23-s + 2.31·25-s − 2·27-s − 3.29·31-s − 2.18·33-s + 24/7·35-s + 0.594·37-s − 0.651·43-s + 28/5·45-s + 2.80·47-s + 49-s + 5.64·51-s + 2.26·53-s + 2.61·55-s − 4.42·57-s + 0.406·59-s − 1.18·61-s + 10/3·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.598551051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.598551051\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T + 51 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T + 86 T^{2} - 456 T^{3} + 2019 T^{4} - 456 p^{2} T^{5} + 86 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T - 85 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 98 T^{2} + 54915 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 48 T + 1514 T^{2} + 35808 T^{3} + 694947 T^{4} + 35808 p^{2} T^{5} + 1514 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 42 T + 1433 T^{2} - 35490 T^{3} + 795972 T^{4} - 35490 p^{2} T^{5} + 1433 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 24 T + 22 T^{2} - 12096 T^{3} - 277629 T^{4} - 12096 p^{2} T^{5} + 22 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 102 T + 6041 T^{2} + 8466 p T^{3} + 9396 p^{2} T^{4} + 8466 p^{3} T^{5} + 6041 p^{4} T^{6} + 102 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 22 T - 2087 T^{2} + 3674 T^{3} + 4073284 T^{4} + 3674 p^{2} T^{5} - 2087 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2476 T^{2} + 6405414 T^{4} - 2476 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 14 T + 3675 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T + 11654 T^{2} - 771672 T^{3} + 42125907 T^{4} - 771672 p^{2} T^{5} + 11654 p^{4} T^{6} - 132 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 120 T + 110 p T^{2} - 354240 T^{3} + 25104819 T^{4} - 354240 p^{2} T^{5} + 110 p^{5} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 6026 T^{2} - 140016 T^{3} + 22586547 T^{4} - 140016 p^{2} T^{5} + 6026 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 72 T + 9218 T^{2} + 539280 T^{3} + 48684147 T^{4} + 539280 p^{2} T^{5} + 9218 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 110 T + 3625 T^{2} - 55330 T^{3} - 2642396 T^{4} - 55330 p^{2} T^{5} + 3625 p^{4} T^{6} + 110 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 156 T + 178 p T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 66 T + 2873 T^{2} + 93786 T^{3} - 18641292 T^{4} + 93786 p^{2} T^{5} + 2873 p^{4} T^{6} + 66 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 10 T - 3695 T^{2} + 86870 T^{3} - 25172156 T^{4} + 86870 p^{2} T^{5} - 3695 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 116 p T^{2} + 107694438 T^{4} - 116 p^{5} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 36 T + 8353 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 36580 T^{2} + 511416774 T^{4} - 36580 p^{4} T^{6} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.151835481077152472907957382424, −7.65419972440289389774311525835, −7.48418616250043892695894158533, −7.25289639055717817922697493637, −7.14497368805639321203491959667, −6.90785231855669327180410414564, −6.42123671748674941584836250427, −6.22123382742639049479108146886, −5.92285178242963670527150351303, −5.83396232702134959754649287498, −5.59350061630147971114709242051, −5.35241684172126701457131833665, −5.29085586208094683692011100207, −4.58708171728656882885678640825, −4.54094877279726719753516454754, −4.39094462496383554117165315713, −3.82836960282743377088542992836, −3.68878024531006319202674696792, −2.89814070708283217346319003471, −2.35070845570825123393360283764, −2.24475340385583045637862320816, −1.65233983664443142631883754336, −1.49375913020863040215864666091, −1.23213891163193174025746937530, −0.30694248599785445817418134248,
0.30694248599785445817418134248, 1.23213891163193174025746937530, 1.49375913020863040215864666091, 1.65233983664443142631883754336, 2.24475340385583045637862320816, 2.35070845570825123393360283764, 2.89814070708283217346319003471, 3.68878024531006319202674696792, 3.82836960282743377088542992836, 4.39094462496383554117165315713, 4.54094877279726719753516454754, 4.58708171728656882885678640825, 5.29085586208094683692011100207, 5.35241684172126701457131833665, 5.59350061630147971114709242051, 5.83396232702134959754649287498, 5.92285178242963670527150351303, 6.22123382742639049479108146886, 6.42123671748674941584836250427, 6.90785231855669327180410414564, 7.14497368805639321203491959667, 7.25289639055717817922697493637, 7.48418616250043892695894158533, 7.65419972440289389774311525835, 8.151835481077152472907957382424
