| L(s) = 1 | − 4·2-s + 8·4-s − 12·5-s + 4·7-s − 8·8-s + 6·9-s + 48·10-s − 16·14-s − 4·16-s − 24·18-s + 52·19-s − 96·20-s + 72·25-s + 32·28-s + 72·29-s − 4·31-s + 32·32-s − 48·35-s + 48·36-s + 68·37-s − 208·38-s + 96·40-s − 60·41-s − 72·45-s + 144·47-s + 8·49-s − 288·50-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2·4-s − 2.39·5-s + 4/7·7-s − 8-s + 2/3·9-s + 24/5·10-s − 8/7·14-s − 1/4·16-s − 4/3·18-s + 2.73·19-s − 4.79·20-s + 2.87·25-s + 8/7·28-s + 2.48·29-s − 0.129·31-s + 32-s − 1.37·35-s + 4/3·36-s + 1.83·37-s − 5.47·38-s + 12/5·40-s − 1.46·41-s − 8/5·45-s + 3.06·47-s + 8/49·49-s − 5.75·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8908374460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8908374460\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 444 T^{3} + 2594 T^{4} + 444 p^{2} T^{5} + 72 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 180 T^{3} + 4034 T^{4} - 180 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 26414 T^{4} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 268 T^{2} + 12198 T^{4} - 268 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T + 1352 T^{2} - 36036 T^{3} + 850274 T^{4} - 36036 p^{2} T^{5} + 1352 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1228 T^{2} + 763878 T^{4} - 1228 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 36 T + 1814 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 1548 T^{3} - 1517566 T^{4} - 1548 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 68 T + 2312 T^{2} - 106284 T^{3} + 4848302 T^{4} - 106284 p^{2} T^{5} + 2312 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 60 T + 1800 T^{2} + 23820 T^{3} - 1333438 T^{4} + 23820 p^{2} T^{5} + 1800 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 5380 T^{2} + 13933734 T^{4} - 5380 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 144 T + 10368 T^{2} - 14544 p T^{3} + 17486 p^{2} T^{4} - 14544 p^{3} T^{5} + 10368 p^{4} T^{6} - 144 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 60 T + 5750 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} + 91128 T^{3} - 24134866 T^{4} + 91128 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 96 T + 6674 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 28 T + 392 T^{2} + 33012 T^{3} - 29346142 T^{4} + 33012 p^{2} T^{5} + 392 p^{4} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 70728 T^{3} + 12984782 T^{4} + 70728 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 92 T + 4232 T^{2} + 532404 T^{3} + 66768974 T^{4} + 532404 p^{2} T^{5} + 4232 p^{4} T^{6} + 92 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 144 T + 11858 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 72 T + 2592 T^{2} + 670824 T^{3} - 89021458 T^{4} + 670824 p^{2} T^{5} + 2592 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 83100 T^{3} + 94919234 T^{4} + 83100 p^{2} T^{5} + 72 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 164 T + 13448 T^{2} - 1429260 T^{3} + 151420814 T^{4} - 1429260 p^{2} T^{5} + 13448 p^{4} T^{6} - 164 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−7.11846999631521134477721575102, −6.94994195876238352412047627420, −6.81316542095079358634302932126, −6.48430804795055567557933416461, −6.13691273627329897032358084516, −5.66732307774343196320616465628, −5.61560138042212646409016162531, −5.26562633322573062815957993919, −5.25517521556772121726288937409, −4.63213910801004976328219874811, −4.59905220651031598906255583930, −4.13007319425211358611338347338, −4.12822541114516897500728299497, −4.07502518039284188908941231113, −3.46126753111715763489982446503, −3.27879807225990424080896429377, −3.02086074172713140816526472065, −2.67149833920406440975749280577, −2.35074224672804575744251775613, −2.14149380697994746117407630618, −1.41929881342642708982597681322, −1.06605481605288875036560337287, −0.894516102222016927445238338430, −0.854215500561437292741241704593, −0.27872729083299149190551868856,
0.27872729083299149190551868856, 0.854215500561437292741241704593, 0.894516102222016927445238338430, 1.06605481605288875036560337287, 1.41929881342642708982597681322, 2.14149380697994746117407630618, 2.35074224672804575744251775613, 2.67149833920406440975749280577, 3.02086074172713140816526472065, 3.27879807225990424080896429377, 3.46126753111715763489982446503, 4.07502518039284188908941231113, 4.12822541114516897500728299497, 4.13007319425211358611338347338, 4.59905220651031598906255583930, 4.63213910801004976328219874811, 5.25517521556772121726288937409, 5.26562633322573062815957993919, 5.61560138042212646409016162531, 5.66732307774343196320616465628, 6.13691273627329897032358084516, 6.48430804795055567557933416461, 6.81316542095079358634302932126, 6.94994195876238352412047627420, 7.11846999631521134477721575102
