| L(s) = 1 | + 4·3-s + 5·4-s + 2·9-s + 20·12-s − 6·13-s + 9·16-s − 8·17-s + 12·23-s + 20·25-s − 12·27-s − 8·29-s + 10·36-s − 24·39-s + 8·43-s + 36·48-s − 32·51-s − 30·52-s + 20·53-s + 12·61-s + 10·64-s − 40·68-s + 48·69-s + 80·75-s − 20·79-s − 17·81-s − 32·87-s + 60·92-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 5/2·4-s + 2/3·9-s + 5.77·12-s − 1.66·13-s + 9/4·16-s − 1.94·17-s + 2.50·23-s + 4·25-s − 2.30·27-s − 1.48·29-s + 5/3·36-s − 3.84·39-s + 1.21·43-s + 5.19·48-s − 4.48·51-s − 4.16·52-s + 2.74·53-s + 1.53·61-s + 5/4·64-s − 4.85·68-s + 5.77·69-s + 9.23·75-s − 2.25·79-s − 1.88·81-s − 3.43·87-s + 6.25·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.195646216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.195646216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + 6 T + 28 T^{2} + 10 p T^{3} + 46 p T^{4} + 10 p^{2} T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 2 | \( 1 - 5 T^{2} + p^{4} T^{4} - 45 T^{6} + 103 T^{8} - 45 p^{2} T^{10} + p^{8} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} \) |
| 3 | \( ( 1 - 2 T + 5 T^{2} - 8 T^{3} + 14 T^{4} - 8 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 - 4 p T^{2} + 203 T^{4} - 56 p^{2} T^{6} + 7648 T^{8} - 56 p^{4} T^{10} + 203 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 36 T^{2} + 552 T^{4} - 6040 T^{6} + 65873 T^{8} - 6040 p^{2} T^{10} + 552 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 4 T + 48 T^{2} + 152 T^{3} + 1177 T^{4} + 152 p T^{5} + 48 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 108 T^{2} + 5632 T^{4} - 184552 T^{6} + 4170841 T^{8} - 184552 p^{2} T^{10} + 5632 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 6 T + 97 T^{2} - 404 T^{3} + 3398 T^{4} - 404 p T^{5} + 97 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 4 T + 53 T^{2} + 140 T^{3} + 2016 T^{4} + 140 p T^{5} + 53 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 168 T^{2} + 14119 T^{4} - 754888 T^{6} + 27862780 T^{8} - 754888 p^{2} T^{10} + 14119 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 - 176 T^{2} + 16039 T^{4} - 966152 T^{6} + 41670748 T^{8} - 966152 p^{2} T^{10} + 16039 p^{4} T^{12} - 176 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 - 196 T^{2} + 20328 T^{4} - 1383052 T^{6} + 66725070 T^{8} - 1383052 p^{2} T^{10} + 20328 p^{4} T^{12} - 196 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 4 T + 106 T^{2} - 672 T^{3} + 5314 T^{4} - 672 p T^{5} + 106 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 180 T^{2} + 18792 T^{4} - 1371592 T^{6} + 73763777 T^{8} - 1371592 p^{2} T^{10} + 18792 p^{4} T^{12} - 180 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 10 T + 4 p T^{2} - 1460 T^{3} + 16941 T^{4} - 1460 p T^{5} + 4 p^{3} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 284 T^{2} + 41392 T^{4} - 3995400 T^{6} + 275818393 T^{8} - 3995400 p^{2} T^{10} + 41392 p^{4} T^{12} - 284 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 6 T + 220 T^{2} - 1004 T^{3} + 19621 T^{4} - 1004 p T^{5} + 220 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 252 T^{2} + 33176 T^{4} - 3237176 T^{6} + 247467313 T^{8} - 3237176 p^{2} T^{10} + 33176 p^{4} T^{12} - 252 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 - 276 T^{2} + 36585 T^{4} - 3341788 T^{6} + 252824924 T^{8} - 3341788 p^{2} T^{10} + 36585 p^{4} T^{12} - 276 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 324 T^{2} + 55307 T^{4} - 6362312 T^{6} + 537626848 T^{8} - 6362312 p^{2} T^{10} + 55307 p^{4} T^{12} - 324 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 10 T + 325 T^{2} + 2310 T^{3} + 38860 T^{4} + 2310 p T^{5} + 325 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 368 T^{2} + 74176 T^{4} - 9998736 T^{6} + 969195550 T^{8} - 9998736 p^{2} T^{10} + 74176 p^{4} T^{12} - 368 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 272 T^{2} + 49727 T^{4} - 6652904 T^{6} + 662082940 T^{8} - 6652904 p^{2} T^{10} + 49727 p^{4} T^{12} - 272 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 672 T^{2} + 205664 T^{4} - 37489952 T^{6} + 4452439678 T^{8} - 37489952 p^{2} T^{10} + 205664 p^{4} T^{12} - 672 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.77679375905218417158290819492, −4.47775075558411272574369073383, −4.30642696289158454565441191315, −4.20014759570961887573140485826, −3.93517832464272092468138988249, −3.79441106287158551614691090714, −3.69820611886200440360573019384, −3.40246319201925240981682139158, −3.36090369820137091405352810323, −3.16043400913848469774021991788, −3.12480244319791597032350341202, −2.93616216430445423900116582820, −2.74027316922378764702108304523, −2.71242691506721898770817309831, −2.40012539105666643596393332657, −2.35348291911943812986449471216, −2.33565029953199214817840055721, −2.31665331123374573648386434864, −2.06680643627822962523039156360, −2.02589112340565463278156527685, −1.27911402945754400666877495035, −1.25797718227731155594466908914, −1.18321035971159136418423054807, −0.866227322488869395308196838982, −0.19957479029904661661591411224,
0.19957479029904661661591411224, 0.866227322488869395308196838982, 1.18321035971159136418423054807, 1.25797718227731155594466908914, 1.27911402945754400666877495035, 2.02589112340565463278156527685, 2.06680643627822962523039156360, 2.31665331123374573648386434864, 2.33565029953199214817840055721, 2.35348291911943812986449471216, 2.40012539105666643596393332657, 2.71242691506721898770817309831, 2.74027316922378764702108304523, 2.93616216430445423900116582820, 3.12480244319791597032350341202, 3.16043400913848469774021991788, 3.36090369820137091405352810323, 3.40246319201925240981682139158, 3.69820611886200440360573019384, 3.79441106287158551614691090714, 3.93517832464272092468138988249, 4.20014759570961887573140485826, 4.30642696289158454565441191315, 4.47775075558411272574369073383, 4.77679375905218417158290819492
Plot not available for L-functions of degree greater than 10.
