| L(s) = 1 | + 4·2-s + 4·4-s − 5·5-s − 16·8-s − 20·10-s + 67·11-s − 82·13-s − 64·16-s + 92·17-s + 43·19-s − 20·20-s + 268·22-s + 148·23-s − 80·25-s − 328·26-s − 154·29-s − 520·31-s − 64·32-s + 368·34-s − 7·37-s + 172·38-s + 80·40-s − 852·41-s − 214·43-s + 268·44-s + 592·46-s − 576·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.447·5-s − 0.707·8-s − 0.632·10-s + 1.83·11-s − 1.74·13-s − 16-s + 1.31·17-s + 0.519·19-s − 0.223·20-s + 2.59·22-s + 1.34·23-s − 0.639·25-s − 2.47·26-s − 0.986·29-s − 3.01·31-s − 0.353·32-s + 1.85·34-s − 0.0311·37-s + 0.734·38-s + 0.316·40-s − 3.24·41-s − 0.758·43-s + 0.918·44-s + 1.89·46-s − 1.78·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5914546154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5914546154\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + p T + 21 p T^{2} - 66 p^{2} T^{3} - 494 p^{2} T^{4} - 66 p^{5} T^{5} + 21 p^{7} T^{6} + p^{10} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 67 T + 1041 T^{2} - 52662 T^{3} + 4142284 T^{4} - 52662 p^{3} T^{5} + 1041 p^{6} T^{6} - 67 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 41 T + 4478 T^{2} + 41 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 92 T + 1902 T^{2} + 17664 p T^{3} - 78413 p^{2} T^{4} + 17664 p^{4} T^{5} + 1902 p^{6} T^{6} - 92 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 43 T - 9305 T^{2} + 110252 T^{3} + 64683544 T^{4} + 110252 p^{3} T^{5} - 9305 p^{6} T^{6} - 43 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 148 T - 2526 T^{2} - 14208 T^{3} + 182283043 T^{4} - 14208 p^{3} T^{5} - 2526 p^{6} T^{6} - 148 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 77 T + 9574 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 520 T + 144563 T^{2} + 34452600 T^{3} + 6891960488 T^{4} + 34452600 p^{3} T^{5} + 144563 p^{6} T^{6} + 520 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 7 T - 74033 T^{2} - 190568 T^{3} + 2919934318 T^{4} - 190568 p^{3} T^{5} - 74033 p^{6} T^{6} + 7 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 426 T + 171106 T^{2} + 426 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 107 T + 86220 T^{2} + 107 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 576 T + 89606 T^{2} + 19885824 T^{3} + 13421113923 T^{4} + 19885824 p^{3} T^{5} + 89606 p^{6} T^{6} + 576 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 243 T - 250441 T^{2} + 2851848 T^{3} + 64828660998 T^{4} + 2851848 p^{3} T^{5} - 250441 p^{6} T^{6} + 243 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7 T - 200565 T^{2} + 1471008 T^{3} - 1944620216 T^{4} + 1471008 p^{3} T^{5} - 200565 p^{6} T^{6} - 7 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 224 T - 410950 T^{2} - 1604736 T^{3} + 149727814859 T^{4} - 1604736 p^{3} T^{5} - 410950 p^{6} T^{6} - 224 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 687 T - 206863 T^{2} + 53109222 T^{3} + 228403689708 T^{4} + 53109222 p^{3} T^{5} - 206863 p^{6} T^{6} + 687 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 472 T + 637018 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 921 T + 270053 T^{2} - 184058166 T^{3} - 147013032042 T^{4} - 184058166 p^{3} T^{5} + 270053 p^{6} T^{6} + 921 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 526 T - 757051 T^{2} - 25063374 T^{3} + 689091996644 T^{4} - 25063374 p^{3} T^{5} - 757051 p^{6} T^{6} - 526 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 221 T + 945628 T^{2} + 221 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 774 T - 367486 T^{2} - 343173024 T^{3} + 14930800239 T^{4} - 343173024 p^{3} T^{5} - 367486 p^{6} T^{6} + 774 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 1953 T + 2366992 T^{2} + 1953 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.75719186558209215686050713171, −6.65171624209624702610408500400, −6.62155424679114290935290741838, −5.96050089426310260449776166766, −5.88371795828572868664824978591, −5.52414333908519000875737740791, −5.41727240921048414782676705073, −5.24123556847711730260899033449, −4.98756530323137534015746258608, −4.84436891232929639537566900991, −4.48282589900028257719801066058, −4.14462611105665257581236173586, −4.03944049474429555707025198139, −3.66142747260451674768901084296, −3.64453363529124792443397353958, −3.11936717356964357497432543143, −3.08918601665691151542842637928, −2.95933247193422677382675170296, −2.45753723302696429280019665467, −1.71194270310842987835749204537, −1.65212421980913176842836742209, −1.64344715268488031470144209665, −1.16898988492149796530960660491, −0.38892006346671671723045187364, −0.10699929065387199215873938850,
0.10699929065387199215873938850, 0.38892006346671671723045187364, 1.16898988492149796530960660491, 1.64344715268488031470144209665, 1.65212421980913176842836742209, 1.71194270310842987835749204537, 2.45753723302696429280019665467, 2.95933247193422677382675170296, 3.08918601665691151542842637928, 3.11936717356964357497432543143, 3.64453363529124792443397353958, 3.66142747260451674768901084296, 4.03944049474429555707025198139, 4.14462611105665257581236173586, 4.48282589900028257719801066058, 4.84436891232929639537566900991, 4.98756530323137534015746258608, 5.24123556847711730260899033449, 5.41727240921048414782676705073, 5.52414333908519000875737740791, 5.88371795828572868664824978591, 5.96050089426310260449776166766, 6.62155424679114290935290741838, 6.65171624209624702610408500400, 6.75719186558209215686050713171
