| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2534805494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2534805494\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 659 T^{2} + p^{6} T^{4} \) |
| 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
−11.62411833615134343538942354176, −11.11101224817046887291525549230, −10.59826071259769249622737241912, −10.49194978002012518055950307025, −10.11224012363320507584594790054, −10.04101611641043452602535908573, −9.908676799935076287744025910980, −9.068926542399521256886882367120, −8.741572613872767076946512710567, −8.264397713952902761381018734402, −8.163339661400542710803417806359, −8.094211193922407281610686539107, −7.63038525208443911588343014653, −7.02002881950550454801392308947, −6.55366399008178928185582293226, −6.09982045050864616557324800905, −6.05589916658781961743436304706, −5.16171461966220405252643999397, −5.06887335527864276188583450795, −4.56474092823162218373314153066, −3.73622577202310628140833981999, −3.33872862581885107678518207610, −2.37557604925372666418076736345, −1.28186826497142456834373264383, −0.41850663403096990490243996388,
0.41850663403096990490243996388, 1.28186826497142456834373264383, 2.37557604925372666418076736345, 3.33872862581885107678518207610, 3.73622577202310628140833981999, 4.56474092823162218373314153066, 5.06887335527864276188583450795, 5.16171461966220405252643999397, 6.05589916658781961743436304706, 6.09982045050864616557324800905, 6.55366399008178928185582293226, 7.02002881950550454801392308947, 7.63038525208443911588343014653, 8.094211193922407281610686539107, 8.163339661400542710803417806359, 8.264397713952902761381018734402, 8.741572613872767076946512710567, 9.068926542399521256886882367120, 9.908676799935076287744025910980, 10.04101611641043452602535908573, 10.11224012363320507584594790054, 10.49194978002012518055950307025, 10.59826071259769249622737241912, 11.11101224817046887291525549230, 11.62411833615134343538942354176
