| L(s) = 1 | − 8·2-s + 3-s + 40·4-s − 8·6-s − 160·8-s − 17·9-s − 10·11-s + 40·12-s − 94·13-s + 560·16-s − 99·17-s + 136·18-s + 246·19-s + 80·22-s + 2·23-s − 160·24-s + 752·26-s + 39·27-s − 98·29-s + 304·31-s − 1.79e3·32-s − 10·33-s + 792·34-s − 680·36-s + 82·37-s − 1.96e3·38-s − 94·39-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 0.192·3-s + 5·4-s − 0.544·6-s − 7.07·8-s − 0.629·9-s − 0.274·11-s + 0.962·12-s − 2.00·13-s + 35/4·16-s − 1.41·17-s + 1.78·18-s + 2.97·19-s + 0.775·22-s + 0.0181·23-s − 1.36·24-s + 5.67·26-s + 0.277·27-s − 0.627·29-s + 1.76·31-s − 9.89·32-s − 0.0527·33-s + 3.99·34-s − 3.14·36-s + 0.364·37-s − 8.40·38-s − 0.385·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.167477875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.167477875\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 - T + 2 p^{2} T^{2} - 74 T^{3} + 284 T^{4} - 74 p^{3} T^{5} + 2 p^{8} T^{6} - p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 10 T + 887 T^{2} + 42279 T^{3} - 1058268 T^{4} + 42279 p^{3} T^{5} + 887 p^{6} T^{6} + 10 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 94 T + 9469 T^{2} + 547643 T^{3} + 31743098 T^{4} + 547643 p^{3} T^{5} + 9469 p^{6} T^{6} + 94 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 99 T + 7907 T^{2} + 24219 p T^{3} + 11623749 T^{4} + 24219 p^{4} T^{5} + 7907 p^{6} T^{6} + 99 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 246 T + 43627 T^{2} - 5128673 T^{3} + 490130304 T^{4} - 5128673 p^{3} T^{5} + 43627 p^{6} T^{6} - 246 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 2 T + 36335 T^{2} - 401169 T^{3} + 600589263 T^{4} - 401169 p^{3} T^{5} + 36335 p^{6} T^{6} - 2 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 98 T + 47147 T^{2} + 10613253 T^{3} + 1071988242 T^{4} + 10613253 p^{3} T^{5} + 47147 p^{6} T^{6} + 98 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 304 T + 112231 T^{2} - 21523619 T^{3} + 4818684785 T^{4} - 21523619 p^{3} T^{5} + 112231 p^{6} T^{6} - 304 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 82 T + 61489 T^{2} - 2368105 T^{3} + 1131558712 T^{4} - 2368105 p^{3} T^{5} + 61489 p^{6} T^{6} - 82 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 352 T + 275417 T^{2} - 68223093 T^{3} + 28548218259 T^{4} - 68223093 p^{3} T^{5} + 275417 p^{6} T^{6} - 352 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 131 T + 218782 T^{2} + 38797580 T^{3} + 22225766680 T^{4} + 38797580 p^{3} T^{5} + 218782 p^{6} T^{6} + 131 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 491 T + 118997 T^{2} + 25437723 T^{3} - 18955187541 T^{4} + 25437723 p^{3} T^{5} + 118997 p^{6} T^{6} - 491 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 140 T + 385925 T^{2} - 100321005 T^{3} + 68698328658 T^{4} - 100321005 p^{3} T^{5} + 385925 p^{6} T^{6} - 140 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 673 T + 856220 T^{2} - 412908738 T^{3} + 267256499478 T^{4} - 412908738 p^{3} T^{5} + 856220 p^{6} T^{6} - 673 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 1425 T + 1404346 T^{2} - 940657748 T^{3} + 519128742930 T^{4} - 940657748 p^{3} T^{5} + 1404346 p^{6} T^{6} - 1425 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 666 T + 1046776 T^{2} - 458997898 T^{3} + 435376596078 T^{4} - 458997898 p^{3} T^{5} + 1046776 p^{6} T^{6} - 666 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 6 T + 1017788 T^{2} + 115021872 T^{3} + 466452598029 T^{4} + 115021872 p^{3} T^{5} + 1017788 p^{6} T^{6} + 6 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 78 T + 330844 T^{2} - 38137306 T^{3} + 326882746518 T^{4} - 38137306 p^{3} T^{5} + 330844 p^{6} T^{6} - 78 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 1744 T + 2829259 T^{2} - 2696615543 T^{3} + 2302296078785 T^{4} - 2696615543 p^{3} T^{5} + 2829259 p^{6} T^{6} - 1744 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 926 T + 1811819 T^{2} + 1321994577 T^{3} + 1501067279118 T^{4} + 1321994577 p^{3} T^{5} + 1811819 p^{6} T^{6} + 926 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 871 T + 2025497 T^{2} + 1288637943 T^{3} + 1956857173959 T^{4} + 1288637943 p^{3} T^{5} + 2025497 p^{6} T^{6} + 871 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 1462 T + 3995479 T^{2} - 3851377603 T^{3} + 5619849626701 T^{4} - 3851377603 p^{3} T^{5} + 3995479 p^{6} T^{6} - 1462 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−6.24970385466971691606488966998, −5.80531778715532955961523182390, −5.57041339634453884994138454095, −5.44454067211893162865101385655, −5.27224047132581184200526865448, −5.07176741686858523027142487956, −4.81736769421226570382827821841, −4.58851213269724466867674178217, −4.33633198600730530277378004078, −3.80180943574076151625521460515, −3.69696730084236623818049205588, −3.64031514801831934553350587564, −3.22471695278858192574118822800, −2.65302560510055173865691771213, −2.62177024108957571664286746398, −2.61072200975962923563369885457, −2.59522304839495093837802306173, −2.14009349440515703237284073828, −1.78091364980536340214215795865, −1.62900717498672196196705528294, −1.17306656102640128908796673640, −0.72028907494649994932929994903, −0.65117797776445062041082661386, −0.64779058378148053204729155396, −0.32533465452306276072102899363,
0.32533465452306276072102899363, 0.64779058378148053204729155396, 0.65117797776445062041082661386, 0.72028907494649994932929994903, 1.17306656102640128908796673640, 1.62900717498672196196705528294, 1.78091364980536340214215795865, 2.14009349440515703237284073828, 2.59522304839495093837802306173, 2.61072200975962923563369885457, 2.62177024108957571664286746398, 2.65302560510055173865691771213, 3.22471695278858192574118822800, 3.64031514801831934553350587564, 3.69696730084236623818049205588, 3.80180943574076151625521460515, 4.33633198600730530277378004078, 4.58851213269724466867674178217, 4.81736769421226570382827821841, 5.07176741686858523027142487956, 5.27224047132581184200526865448, 5.44454067211893162865101385655, 5.57041339634453884994138454095, 5.80531778715532955961523182390, 6.24970385466971691606488966998
