| L(s) = 1 | − 4·3-s + 4·5-s + 8·9-s + 8·11-s − 12·13-s − 16·15-s + 8·17-s − 4·19-s + 8·25-s − 12·27-s − 8·31-s − 32·33-s + 8·37-s + 48·39-s + 24·43-s + 32·45-s − 40·47-s − 4·49-s − 32·51-s − 16·53-s + 32·55-s + 16·57-s + 52·59-s + 20·61-s − 48·65-s − 32·67-s − 32·75-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.78·5-s + 8/3·9-s + 2.41·11-s − 3.32·13-s − 4.13·15-s + 1.94·17-s − 0.917·19-s + 8/5·25-s − 2.30·27-s − 1.43·31-s − 5.57·33-s + 1.31·37-s + 7.68·39-s + 3.65·43-s + 4.77·45-s − 5.83·47-s − 4/7·49-s − 4.48·51-s − 2.19·53-s + 4.31·55-s + 2.11·57-s + 6.76·59-s + 2.56·61-s − 5.95·65-s − 3.90·67-s − 3.69·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.044910192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.044910192\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| good | 3 | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + 4 p T^{4} + 20 T^{5} + 56 T^{6} + 140 T^{7} + 298 T^{8} + 140 p T^{9} + 56 p^{2} T^{10} + 20 p^{3} T^{11} + 4 p^{5} T^{12} + 4 p^{6} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - 4 T + 8 T^{2} + 4 T^{3} + 12 T^{4} - 172 T^{5} + 24 p^{2} T^{6} - 84 p T^{7} - 86 T^{8} - 84 p^{2} T^{9} + 24 p^{4} T^{10} - 172 p^{3} T^{11} + 12 p^{4} T^{12} + 4 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 8 T + 32 T^{2} - 56 T^{3} + 4 T^{4} + 24 p T^{5} - 672 T^{6} + 2680 T^{7} - 8346 T^{8} + 2680 p T^{9} - 672 p^{2} T^{10} + 24 p^{4} T^{11} + 4 p^{4} T^{12} - 56 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( 1 + 12 T + 72 T^{2} + 28 p T^{3} + 1724 T^{4} + 6892 T^{5} + 24824 T^{6} + 88540 T^{7} + 316650 T^{8} + 88540 p T^{9} + 24824 p^{2} T^{10} + 6892 p^{3} T^{11} + 1724 p^{4} T^{12} + 28 p^{6} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 4 T + 64 T^{2} - 188 T^{3} + 1606 T^{4} - 188 p T^{5} + 64 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 4 T + 8 T^{2} + 100 T^{3} + 220 T^{4} + 268 T^{5} + 4312 T^{6} + 19452 T^{7} + 119274 T^{8} + 19452 p T^{9} + 4312 p^{2} T^{10} + 268 p^{3} T^{11} + 220 p^{4} T^{12} + 100 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 - 128 T^{2} + 7876 T^{4} - 307584 T^{6} + 8371142 T^{8} - 307584 p^{2} T^{10} + 7876 p^{4} T^{12} - 128 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( 1 + 128 T^{3} + 28 T^{4} - 4480 T^{5} + 8192 T^{6} - 22784 T^{7} - 1337754 T^{8} - 22784 p T^{9} + 8192 p^{2} T^{10} - 4480 p^{3} T^{11} + 28 p^{4} T^{12} + 128 p^{5} T^{13} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 4 T + 80 T^{2} + 404 T^{3} + 3046 T^{4} + 404 p T^{5} + 80 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 8 T + 32 T^{2} - 344 T^{3} + 1212 T^{4} + 7768 T^{5} - 41760 T^{6} + 554952 T^{7} - 6787994 T^{8} + 554952 p T^{9} - 41760 p^{2} T^{10} + 7768 p^{3} T^{11} + 1212 p^{4} T^{12} - 344 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( 1 - 144 T^{2} + 10540 T^{4} - 527216 T^{6} + 22055462 T^{8} - 527216 p^{2} T^{10} + 10540 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( 1 - 24 T + 288 T^{2} - 2696 T^{3} + 22148 T^{4} - 155528 T^{5} + 988256 T^{6} - 5986904 T^{7} + 37147494 T^{8} - 5986904 p T^{9} + 988256 p^{2} T^{10} - 155528 p^{3} T^{11} + 22148 p^{4} T^{12} - 2696 p^{5} T^{13} + 288 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 20 T + 328 T^{2} + 3220 T^{3} + 26806 T^{4} + 3220 p T^{5} + 328 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 + 16 T + 128 T^{2} + 1328 T^{3} + 16444 T^{4} + 127696 T^{5} + 820096 T^{6} + 7198320 T^{7} + 62713830 T^{8} + 7198320 p T^{9} + 820096 p^{2} T^{10} + 127696 p^{3} T^{11} + 16444 p^{4} T^{12} + 1328 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 - 52 T + 1352 T^{2} - 24452 T^{3} + 350812 T^{4} - 4208044 T^{5} + 43470616 T^{6} - 395995788 T^{7} + 3217667370 T^{8} - 395995788 p T^{9} + 43470616 p^{2} T^{10} - 4208044 p^{3} T^{11} + 350812 p^{4} T^{12} - 24452 p^{5} T^{13} + 1352 p^{6} T^{14} - 52 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 - 20 T + 200 T^{2} - 2300 T^{3} + 15500 T^{4} + 14420 T^{5} - 743400 T^{6} + 12216140 T^{7} - 143039766 T^{8} + 12216140 p T^{9} - 743400 p^{2} T^{10} + 14420 p^{3} T^{11} + 15500 p^{4} T^{12} - 2300 p^{5} T^{13} + 200 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 + 32 T + 512 T^{2} + 5984 T^{3} + 54884 T^{4} + 388192 T^{5} + 2225664 T^{6} + 10093600 T^{7} + 49302630 T^{8} + 10093600 p T^{9} + 2225664 p^{2} T^{10} + 388192 p^{3} T^{11} + 54884 p^{4} T^{12} + 5984 p^{5} T^{13} + 512 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( 1 - 184 T^{2} + 21884 T^{4} - 1920520 T^{6} + 138804294 T^{8} - 1920520 p^{2} T^{10} + 21884 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 216 T^{2} + 32924 T^{4} - 3643752 T^{6} + 294370438 T^{8} - 3643752 p^{2} T^{10} + 32924 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 28 T^{2} + 128 T^{3} + 11030 T^{4} + 128 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 12 T + 72 T^{2} - 980 T^{3} + 9804 T^{4} - 39084 T^{5} + 243320 T^{6} - 3448132 T^{7} + 48827050 T^{8} - 3448132 p T^{9} + 243320 p^{2} T^{10} - 39084 p^{3} T^{11} + 9804 p^{4} T^{12} - 980 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( 1 - 136 T^{2} + 10236 T^{4} - 1385144 T^{6} + 175424582 T^{8} - 1385144 p^{2} T^{10} + 10236 p^{4} T^{12} - 136 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 36 T + 8 p T^{2} - 11436 T^{3} + 128662 T^{4} - 11436 p T^{5} + 8 p^{3} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.11274339599840106204677964284, −3.72775755076277089654546466737, −3.58996105679479375653234326601, −3.53722538114609763713137311634, −3.42693752367482364566478081735, −3.40360842271398285371773633796, −3.22878417130067718862880823837, −3.00606595206472097531870953627, −2.78798807466425500236249501246, −2.72215824581238446567661253217, −2.63159619314241931827351428860, −2.44464245034341192700562490956, −2.17212234542624347416894444015, −2.06247234270350055687480483675, −1.99452108800321007239102896319, −1.97340435712122980588859007371, −1.84490795149228609450499035130, −1.59571953658291757385419142761, −1.25118480229709264252084778326, −1.24064766997111045099347056619, −1.14491674660872516237089545841, −0.805338175732221456827006526963, −0.56452228844791564478928001879, −0.52577393199466286498128115805, −0.21797393171456782389087024663,
0.21797393171456782389087024663, 0.52577393199466286498128115805, 0.56452228844791564478928001879, 0.805338175732221456827006526963, 1.14491674660872516237089545841, 1.24064766997111045099347056619, 1.25118480229709264252084778326, 1.59571953658291757385419142761, 1.84490795149228609450499035130, 1.97340435712122980588859007371, 1.99452108800321007239102896319, 2.06247234270350055687480483675, 2.17212234542624347416894444015, 2.44464245034341192700562490956, 2.63159619314241931827351428860, 2.72215824581238446567661253217, 2.78798807466425500236249501246, 3.00606595206472097531870953627, 3.22878417130067718862880823837, 3.40360842271398285371773633796, 3.42693752367482364566478081735, 3.53722538114609763713137311634, 3.58996105679479375653234326601, 3.72775755076277089654546466737, 4.11274339599840106204677964284
Plot not available for L-functions of degree greater than 10.
