L(s) = 1 | − 8·2-s + 36·4-s + 2·5-s − 120·8-s + 5·9-s − 16·10-s − 6·11-s + 10·13-s + 330·16-s − 40·18-s + 72·20-s + 48·22-s + 12·23-s − 19·25-s − 80·26-s − 14·29-s − 792·32-s + 180·36-s − 240·40-s + 22·41-s − 216·44-s + 10·45-s − 96·46-s + 24·47-s + 33·49-s + 152·50-s + 360·52-s + ⋯ |
L(s) = 1 | − 5.65·2-s + 18·4-s + 0.894·5-s − 42.4·8-s + 5/3·9-s − 5.05·10-s − 1.80·11-s + 2.77·13-s + 82.5·16-s − 9.42·18-s + 16.0·20-s + 10.2·22-s + 2.50·23-s − 3.79·25-s − 15.6·26-s − 2.59·29-s − 140.·32-s + 30·36-s − 37.9·40-s + 3.43·41-s − 32.5·44-s + 1.49·45-s − 14.1·46-s + 3.50·47-s + 33/7·49-s + 21.4·50-s + 49.9·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2640709822\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2640709822\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 + T )^{8} \) |
| 11 | \( 1 + 6 T + 10 T^{2} - 74 T^{3} - 406 T^{4} - 74 p T^{5} + 10 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | \( 1 + 64 T^{3} + 302 T^{4} + 64 p T^{5} + p^{4} T^{8} \) |
good | 3 | \( 1 - 5 T^{2} + 14 T^{4} + 4 p T^{6} - 91 T^{8} + 4 p^{3} T^{10} + 14 p^{4} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( ( 1 - T + 11 T^{2} + T^{3} + 54 T^{4} + p T^{5} + 11 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 - 33 T^{2} + 578 T^{4} - 956 p T^{6} + 54925 T^{8} - 956 p^{3} T^{10} + 578 p^{4} T^{12} - 33 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 5 T + 36 T^{2} - 116 T^{3} + 597 T^{4} - 116 p T^{5} + 36 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 98 T^{2} + 4581 T^{4} - 133986 T^{6} + 2704892 T^{8} - 133986 p^{2} T^{10} + 4581 p^{4} T^{12} - 98 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 6 T + 85 T^{2} - 370 T^{3} + 2900 T^{4} - 370 p T^{5} + 85 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 7 T + 128 T^{2} + 608 T^{3} + 5733 T^{4} + 608 p T^{5} + 128 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 127 T^{2} + 8631 T^{4} - 399369 T^{6} + 13987188 T^{8} - 399369 p^{2} T^{10} + 8631 p^{4} T^{12} - 127 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 - 144 T^{2} + 11708 T^{4} - 658160 T^{6} + 27671398 T^{8} - 658160 p^{2} T^{10} + 11708 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 11 T + 143 T^{2} - 1141 T^{3} + 8814 T^{4} - 1141 p T^{5} + 143 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 199 T^{2} + 19035 T^{4} - 1182645 T^{6} + 56284224 T^{8} - 1182645 p^{2} T^{10} + 19035 p^{4} T^{12} - 199 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 12 T + 160 T^{2} - 1340 T^{3} + 11390 T^{4} - 1340 p T^{5} + 160 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 158 T^{2} + 13293 T^{4} - 744102 T^{6} + 37825628 T^{8} - 744102 p^{2} T^{10} + 13293 p^{4} T^{12} - 158 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 182 T^{2} + 18757 T^{4} - 1506046 T^{6} + 99108428 T^{8} - 1506046 p^{2} T^{10} + 18757 p^{4} T^{12} - 182 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 200 T^{2} + 24432 T^{4} - 2187336 T^{6} + 151865246 T^{8} - 2187336 p^{2} T^{10} + 24432 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( 1 - 277 T^{2} + 38430 T^{4} - 3725244 T^{6} + 281073741 T^{8} - 3725244 p^{2} T^{10} + 38430 p^{4} T^{12} - 277 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 - 255 T^{2} + 39227 T^{4} - 4232957 T^{6} + 342067408 T^{8} - 4232957 p^{2} T^{10} + 39227 p^{4} T^{12} - 255 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 402 T^{2} + 80837 T^{4} - 10258466 T^{6} + 894740476 T^{8} - 10258466 p^{2} T^{10} + 80837 p^{4} T^{12} - 402 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 6 T + 154 T^{2} - 82 T^{3} + 8650 T^{4} - 82 p T^{5} + 154 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 399 T^{2} + 77031 T^{4} - 9924113 T^{6} + 947476004 T^{8} - 9924113 p^{2} T^{10} + 77031 p^{4} T^{12} - 399 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 144 T^{2} + 25820 T^{4} - 2230640 T^{6} + 250392070 T^{8} - 2230640 p^{2} T^{10} + 25820 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 712 T^{2} + 227664 T^{4} - 42629640 T^{6} + 5108602878 T^{8} - 42629640 p^{2} T^{10} + 227664 p^{4} T^{12} - 712 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−5.19779152037793208245951230545, −4.86539884503127954528074462858, −4.76152490521213774089046404982, −4.36279613019512073632240397851, −4.25810879321570677501817729954, −3.93663955112346099773278337424, −3.91810145506553482724426769677, −3.91536457535726972198681858026, −3.87504259152179511301721966887, −3.42112297213876411598424955529, −3.40421306270928754850830862151, −2.94460480807824333099001367892, −2.83482269006274851332282380473, −2.67255709542641031065644977930, −2.55249520937379824189846781373, −2.25528075474567765077414851654, −2.16982713424575487610306520557, −2.15653792522882343340518540329, −1.86191885680465912747821302604, −1.58282879084724722008541701956, −1.27603212719977215711898238829, −1.20113393568195663340293084588, −1.03614327660105366567684803460, −0.74175404979455627038696214270, −0.34230758851287178624933307412,
0.34230758851287178624933307412, 0.74175404979455627038696214270, 1.03614327660105366567684803460, 1.20113393568195663340293084588, 1.27603212719977215711898238829, 1.58282879084724722008541701956, 1.86191885680465912747821302604, 2.15653792522882343340518540329, 2.16982713424575487610306520557, 2.25528075474567765077414851654, 2.55249520937379824189846781373, 2.67255709542641031065644977930, 2.83482269006274851332282380473, 2.94460480807824333099001367892, 3.40421306270928754850830862151, 3.42112297213876411598424955529, 3.87504259152179511301721966887, 3.91536457535726972198681858026, 3.91810145506553482724426769677, 3.93663955112346099773278337424, 4.25810879321570677501817729954, 4.36279613019512073632240397851, 4.76152490521213774089046404982, 4.86539884503127954528074462858, 5.19779152037793208245951230545
Plot not available for L-functions of degree greater than 10.