| L(s) = 1 | − 2·2-s + 8·3-s + 3·4-s − 16·6-s − 2·8-s + 36·9-s + 24·12-s + 3·16-s − 72·18-s + 12·19-s − 16·24-s + 18·25-s + 120·27-s − 16·29-s − 12·31-s − 6·32-s + 108·36-s − 12·37-s − 24·38-s − 8·47-s + 24·48-s − 36·50-s + 8·53-s − 240·54-s + 96·57-s + 32·58-s + 28·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4.61·3-s + 3/2·4-s − 6.53·6-s − 0.707·8-s + 12·9-s + 6.92·12-s + 3/4·16-s − 16.9·18-s + 2.75·19-s − 3.26·24-s + 18/5·25-s + 23.0·27-s − 2.97·29-s − 2.15·31-s − 1.06·32-s + 18·36-s − 1.97·37-s − 3.89·38-s − 1.16·47-s + 3.46·48-s − 5.09·50-s + 1.09·53-s − 32.6·54-s + 12.7·57-s + 4.20·58-s + 3.64·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(35.90720452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(35.90720452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T + T^{2} - p T^{3} - 3 p T^{4} - p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | \( ( 1 - T )^{8} \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 18 T^{2} + 153 T^{4} - 906 T^{6} + 4676 T^{8} - 906 p^{2} T^{10} + 153 p^{4} T^{12} - 18 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 10 T + 25 T^{2} + 94 T^{3} - 684 T^{4} + 94 p T^{5} + 25 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )( 1 + 10 T + 25 T^{2} - 94 T^{3} - 684 T^{4} - 94 p T^{5} + 25 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 13 | \( 1 - 66 T^{2} + 2241 T^{4} - 49362 T^{6} + 759092 T^{8} - 49362 p^{2} T^{10} + 2241 p^{4} T^{12} - 66 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 80 T^{2} + 3228 T^{4} - 87472 T^{6} + 1728326 T^{8} - 87472 p^{2} T^{10} + 3228 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 6 T + 69 T^{2} - 282 T^{3} + 1904 T^{4} - 282 p T^{5} + 69 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 104 T^{2} + 5628 T^{4} - 203992 T^{6} + 5416646 T^{8} - 203992 p^{2} T^{10} + 5628 p^{4} T^{12} - 104 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 8 T + 71 T^{2} + 344 T^{3} + 1924 T^{4} + 344 p T^{5} + 71 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 6 T + 40 T^{2} + 288 T^{3} + 2601 T^{4} + 288 p T^{5} + 40 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 6 T + 105 T^{2} + 306 T^{3} + 4436 T^{4} + 306 p T^{5} + 105 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 120 T^{2} + 9948 T^{4} - 607176 T^{6} + 27583238 T^{8} - 607176 p^{2} T^{10} + 9948 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( 1 - 210 T^{2} + 23793 T^{4} - 1729746 T^{6} + 88400276 T^{8} - 1729746 p^{2} T^{10} + 23793 p^{4} T^{12} - 210 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 4 T + 160 T^{2} + 548 T^{3} + 10686 T^{4} + 548 p T^{5} + 160 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4 T + 151 T^{2} - 752 T^{3} + 10380 T^{4} - 752 p T^{5} + 151 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 14 T + 209 T^{2} - 2018 T^{3} + 18892 T^{4} - 2018 p T^{5} + 209 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( 1 - 216 T^{2} + 27420 T^{4} - 2531496 T^{6} + 176195558 T^{8} - 2531496 p^{2} T^{10} + 27420 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( 1 - 282 T^{2} + 45393 T^{4} - 4904034 T^{6} + 383059316 T^{8} - 4904034 p^{2} T^{10} + 45393 p^{4} T^{12} - 282 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 - 288 T^{2} + 41724 T^{4} - 4336608 T^{6} + 350671046 T^{8} - 4336608 p^{2} T^{10} + 41724 p^{4} T^{12} - 288 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 370 T^{2} + 70161 T^{4} - 8626802 T^{6} + 743765828 T^{8} - 8626802 p^{2} T^{10} + 70161 p^{4} T^{12} - 370 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( 1 - 228 T^{2} + 23202 T^{4} - 2215104 T^{6} + 207545771 T^{8} - 2215104 p^{2} T^{10} + 23202 p^{4} T^{12} - 228 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 2 T + 229 T^{2} - 802 T^{3} + 24432 T^{4} - 802 p T^{5} + 229 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 528 T^{2} + 131868 T^{4} - 20568048 T^{6} + 2191026566 T^{8} - 20568048 p^{2} T^{10} + 131868 p^{4} T^{12} - 528 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 594 T^{2} + 165777 T^{4} - 28537554 T^{6} + 3324136868 T^{8} - 28537554 p^{2} T^{10} + 165777 p^{4} T^{12} - 594 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.52636982754497847590296640885, −4.49044958717648577296240613735, −4.46073526954832380222010472551, −4.01478615756406289244188618418, −3.96858590985942636688055683474, −3.84629860290117885944706242909, −3.63863413530997206586353180598, −3.45461396839785190327207004740, −3.40030963283269115989687941772, −3.28816820824642996789628489241, −3.22026176575951212432392716558, −3.18923047777468384035385363332, −2.92885874166160792111687522660, −2.75096565894104182827875493047, −2.53335899746608145265995416723, −2.51781690064717477487020973278, −2.01903806498172212038453795152, −1.97048130834686713745662045158, −1.93056326903974806631155284945, −1.83576816117743312038232443218, −1.61599471203690492856346109821, −1.45527940591372810576326316697, −0.953702447414658018111453212768, −0.807695450707891344211926856982, −0.75265222924768984434033176839,
0.75265222924768984434033176839, 0.807695450707891344211926856982, 0.953702447414658018111453212768, 1.45527940591372810576326316697, 1.61599471203690492856346109821, 1.83576816117743312038232443218, 1.93056326903974806631155284945, 1.97048130834686713745662045158, 2.01903806498172212038453795152, 2.51781690064717477487020973278, 2.53335899746608145265995416723, 2.75096565894104182827875493047, 2.92885874166160792111687522660, 3.18923047777468384035385363332, 3.22026176575951212432392716558, 3.28816820824642996789628489241, 3.40030963283269115989687941772, 3.45461396839785190327207004740, 3.63863413530997206586353180598, 3.84629860290117885944706242909, 3.96858590985942636688055683474, 4.01478615756406289244188618418, 4.46073526954832380222010472551, 4.49044958717648577296240613735, 4.52636982754497847590296640885
Plot not available for L-functions of degree greater than 10.
