| L(s) = 1 | − 4·2-s + 6·3-s + 4·4-s + 5·5-s − 24·6-s + 16·8-s + 9·9-s − 20·10-s − 67·11-s + 24·12-s − 82·13-s + 30·15-s − 64·16-s − 92·17-s − 36·18-s + 43·19-s + 20·20-s + 268·22-s − 148·23-s + 96·24-s − 80·25-s + 328·26-s − 54·27-s + 154·29-s − 120·30-s − 520·31-s + 64·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 1.63·6-s + 0.707·8-s + 1/3·9-s − 0.632·10-s − 1.83·11-s + 0.577·12-s − 1.74·13-s + 0.516·15-s − 16-s − 1.31·17-s − 0.471·18-s + 0.519·19-s + 0.223·20-s + 2.59·22-s − 1.34·23-s + 0.816·24-s − 0.639·25-s + 2.47·26-s − 0.384·27-s + 0.986·29-s − 0.730·30-s − 3.01·31-s + 0.353·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.09130521029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09130521029\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 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
−8.018501530828628551448918060774, −7.936480338683790001176559096966, −7.55565003644032229753366525302, −7.52479019802730056949740865507, −7.46122620358906704236580464815, −7.00465529752423820732720255064, −6.76791457889360561633335853008, −6.33072367385129767772116084212, −5.93609403693294786822780329581, −5.78023253182408109812016082003, −5.20767890743804762396596558351, −5.20122457338249944588479538557, −5.12771536620029890485343004511, −4.31237766489417542031878253864, −4.25416374499708619914777383916, −3.83659349463606102453932133079, −3.74041758720168146754045239081, −2.72249058556384181380223808557, −2.70281736935727476984077404219, −2.60103290262005161987481281344, −2.16788358256481329236877733480, −1.82244994494784831939437671044, −1.34335788745930428083131141337, −0.55162803863153112264212906285, −0.089878875687861273206985580797,
0.089878875687861273206985580797, 0.55162803863153112264212906285, 1.34335788745930428083131141337, 1.82244994494784831939437671044, 2.16788358256481329236877733480, 2.60103290262005161987481281344, 2.70281736935727476984077404219, 2.72249058556384181380223808557, 3.74041758720168146754045239081, 3.83659349463606102453932133079, 4.25416374499708619914777383916, 4.31237766489417542031878253864, 5.12771536620029890485343004511, 5.20122457338249944588479538557, 5.20767890743804762396596558351, 5.78023253182408109812016082003, 5.93609403693294786822780329581, 6.33072367385129767772116084212, 6.76791457889360561633335853008, 7.00465529752423820732720255064, 7.46122620358906704236580464815, 7.52479019802730056949740865507, 7.55565003644032229753366525302, 7.936480338683790001176559096966, 8.018501530828628551448918060774
