| L(s) = 1 | − 4·3-s − 2·7-s + 6·9-s + 6·11-s − 6·19-s + 8·21-s − 9·25-s + 16·29-s − 6·31-s − 24·33-s − 6·37-s − 4·47-s + 4·49-s + 4·53-s + 24·57-s − 14·59-s − 12·61-s − 12·63-s + 42·67-s − 18·73-s + 36·75-s − 12·77-s + 6·79-s − 15·81-s + 4·83-s − 64·87-s + 24·93-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 0.755·7-s + 2·9-s + 1.80·11-s − 1.37·19-s + 1.74·21-s − 9/5·25-s + 2.97·29-s − 1.07·31-s − 4.17·33-s − 0.986·37-s − 0.583·47-s + 4/7·49-s + 0.549·53-s + 3.17·57-s − 1.82·59-s − 1.53·61-s − 1.51·63-s + 5.13·67-s − 2.10·73-s + 4.15·75-s − 1.36·77-s + 0.675·79-s − 5/3·81-s + 0.439·83-s − 6.86·87-s + 2.48·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.166359912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.166359912\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T + T^{2} )^{4} \) |
| 7 | \( 1 + 2 T + 16 T^{3} + 65 T^{4} + 16 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 5 | \( 1 + 9 T^{2} + 9 p T^{4} - 96 T^{5} + 66 T^{6} - 864 T^{7} - 394 T^{8} - 864 p T^{9} + 66 p^{2} T^{10} - 96 p^{3} T^{11} + 9 p^{5} T^{12} + 9 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 8 T + 34 T^{2} - 112 T^{3} + 339 T^{4} - 112 p T^{5} + 34 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )( 1 + 2 T + 25 T^{2} + 58 T^{3} + 372 T^{4} + 58 p T^{5} + 25 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 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 | \( 1 + 40 T^{2} + 786 T^{4} + 1104 T^{5} + 10208 T^{6} + 39072 T^{7} + 129731 T^{8} + 39072 p T^{9} + 10208 p^{2} T^{10} + 1104 p^{3} T^{11} + 786 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 + 6 T - 33 T^{2} - 150 T^{3} + 1165 T^{4} + 1968 T^{5} - 34182 T^{6} - 11868 T^{7} + 759066 T^{8} - 11868 p T^{9} - 34182 p^{2} T^{10} + 1968 p^{3} T^{11} + 1165 p^{4} T^{12} - 150 p^{5} T^{13} - 33 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 + 52 T^{2} + 54 p T^{4} - 2784 T^{5} + 24080 T^{6} - 140352 T^{7} + 497843 T^{8} - 140352 p T^{9} + 24080 p^{2} T^{10} - 2784 p^{3} T^{11} + 54 p^{5} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} \) |
| 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 | \( 1 + 6 T - 4 T^{2} - 336 T^{3} - 2729 T^{4} - 10764 T^{5} + 2216 T^{6} + 444234 T^{7} + 3877768 T^{8} + 444234 p T^{9} + 2216 p^{2} T^{10} - 10764 p^{3} T^{11} - 2729 p^{4} T^{12} - 336 p^{5} T^{13} - 4 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 + 6 T - 69 T^{2} + 18 T^{3} + 4753 T^{4} - 9780 T^{5} - 152586 T^{6} + 146184 T^{7} + 2893194 T^{8} + 146184 p T^{9} - 152586 p^{2} T^{10} - 9780 p^{3} T^{11} + 4753 p^{4} T^{12} + 18 p^{5} T^{13} - 69 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 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 | \( 1 + 4 T - 144 T^{2} - 456 T^{3} + 12722 T^{4} + 27948 T^{5} - 805600 T^{6} - 556940 T^{7} + 41968563 T^{8} - 556940 p T^{9} - 805600 p^{2} T^{10} + 27948 p^{3} T^{11} + 12722 p^{4} T^{12} - 456 p^{5} T^{13} - 144 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 - 4 T - 135 T^{2} + 900 T^{3} + 9413 T^{4} - 70368 T^{5} - 312982 T^{6} + 1938608 T^{7} + 10598262 T^{8} + 1938608 p T^{9} - 312982 p^{2} T^{10} - 70368 p^{3} T^{11} + 9413 p^{4} T^{12} + 900 p^{5} T^{13} - 135 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 14 T - 13 T^{2} - 1110 T^{3} - 3463 T^{4} + 11848 T^{5} - 87914 T^{6} + 1416852 T^{7} + 32978194 T^{8} + 1416852 p T^{9} - 87914 p^{2} T^{10} + 11848 p^{3} T^{11} - 3463 p^{4} T^{12} - 1110 p^{5} T^{13} - 13 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 12 T + 180 T^{2} + 1584 T^{3} + 12426 T^{4} + 46860 T^{5} + 10032 T^{6} - 3576756 T^{7} - 33274477 T^{8} - 3576756 p T^{9} + 10032 p^{2} T^{10} + 46860 p^{3} T^{11} + 12426 p^{4} T^{12} + 1584 p^{5} T^{13} + 180 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 - 42 T + 1023 T^{2} - 18270 T^{3} + 261141 T^{4} - 3133152 T^{5} + 32837970 T^{6} - 307895844 T^{7} + 2631022010 T^{8} - 307895844 p T^{9} + 32837970 p^{2} T^{10} - 3133152 p^{3} T^{11} + 261141 p^{4} T^{12} - 18270 p^{5} T^{13} + 1023 p^{6} T^{14} - 42 p^{7} T^{15} + p^{8} T^{16} \) |
| 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 | \( 1 + 18 T + 347 T^{2} + 4302 T^{3} + 50601 T^{4} + 463140 T^{5} + 4384558 T^{6} + 34967736 T^{7} + 313616978 T^{8} + 34967736 p T^{9} + 4384558 p^{2} T^{10} + 463140 p^{3} T^{11} + 50601 p^{4} T^{12} + 4302 p^{5} T^{13} + 347 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 - 6 T + 132 T^{2} - 720 T^{3} + 9999 T^{4} - 52644 T^{5} - 314880 T^{6} - 1496442 T^{7} - 48671848 T^{8} - 1496442 p T^{9} - 314880 p^{2} T^{10} - 52644 p^{3} T^{11} + 9999 p^{4} T^{12} - 720 p^{5} T^{13} + 132 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 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 | \( 1 + 264 T^{2} + 38610 T^{4} + 50400 T^{5} + 4052064 T^{6} + 8952768 T^{7} + 353995811 T^{8} + 8952768 p T^{9} + 4052064 p^{2} T^{10} + 50400 p^{3} T^{11} + 38610 p^{4} T^{12} + 264 p^{6} T^{14} + p^{8} T^{16} \) |
| 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
−4.14060401552591275900189937078, −4.10460365184220301708107000958, −4.08540545459923934492897625832, −3.59861100399508950088159468972, −3.54815978398820083207255621740, −3.35980829258018654199859940335, −3.32522301713160558345350510906, −3.29142269926349750503193249207, −3.19992885630399411689697146216, −2.95884435745703627955003100028, −2.79501160281235414252442377383, −2.56015085855146528549165359988, −2.44354951110861199444382502010, −2.36321210396173223734391396073, −2.08324948047265111741992310647, −1.89007662242200476272781832417, −1.76036016625971485532555233889, −1.63551911552543305818212823807, −1.59659212990962645742074197661, −1.41517597615019151410395243076, −0.860724215737522171079632256465, −0.78788609897453614861270177400, −0.60955976387646397949038975039, −0.57383277588557713080498083332, −0.19918339917908269558996081354,
0.19918339917908269558996081354, 0.57383277588557713080498083332, 0.60955976387646397949038975039, 0.78788609897453614861270177400, 0.860724215737522171079632256465, 1.41517597615019151410395243076, 1.59659212990962645742074197661, 1.63551911552543305818212823807, 1.76036016625971485532555233889, 1.89007662242200476272781832417, 2.08324948047265111741992310647, 2.36321210396173223734391396073, 2.44354951110861199444382502010, 2.56015085855146528549165359988, 2.79501160281235414252442377383, 2.95884435745703627955003100028, 3.19992885630399411689697146216, 3.29142269926349750503193249207, 3.32522301713160558345350510906, 3.35980829258018654199859940335, 3.54815978398820083207255621740, 3.59861100399508950088159468972, 4.08540545459923934492897625832, 4.10460365184220301708107000958, 4.14060401552591275900189937078
Plot not available for L-functions of degree greater than 10.
