| L(s) = 1 | − 8·2-s + 4·3-s + 36·4-s − 32·6-s − 120·8-s + 6·9-s + 144·12-s + 330·16-s + 16·17-s − 48·18-s − 480·24-s + 8·27-s − 8·29-s + 24·31-s − 792·32-s − 128·34-s + 216·36-s + 24·37-s + 1.32e3·48-s + 28·49-s + 64·51-s − 64·54-s + 64·58-s − 192·62-s + 1.71e3·64-s + 8·67-s + 576·68-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 2.30·3-s + 18·4-s − 13.0·6-s − 42.4·8-s + 2·9-s + 41.5·12-s + 82.5·16-s + 3.88·17-s − 11.3·18-s − 97.9·24-s + 1.53·27-s − 1.48·29-s + 4.31·31-s − 140.·32-s − 21.9·34-s + 36·36-s + 3.94·37-s + 190.·48-s + 4·49-s + 8.96·51-s − 8.70·54-s + 8.40·58-s − 24.3·62-s + 214.5·64-s + 0.977·67-s + 69.8·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.278842049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.278842049\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 3 | \( 1 - 4 T + 10 T^{2} - 8 p T^{3} + 16 p T^{4} - 8 p^{2} T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 14 T^{4} - 192 T^{6} + 51 T^{8} - 192 p^{2} T^{10} + 14 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( 1 - 4 p T^{2} + 452 T^{4} - 5004 T^{6} + 40490 T^{8} - 5004 p^{2} T^{10} + 452 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 36 T^{2} + 418 T^{4} + 2592 T^{6} - 100725 T^{8} + 2592 p^{2} T^{10} + 418 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 8 T + 60 T^{2} - 312 T^{3} + 1363 T^{4} - 312 p T^{5} + 60 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 84 T^{2} + 3844 T^{4} - 118404 T^{6} + 2616138 T^{8} - 118404 p^{2} T^{10} + 3844 p^{4} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 60 T^{2} + 2612 T^{4} - 87180 T^{6} + 2186346 T^{8} - 87180 p^{2} T^{10} + 2612 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 4 T + 78 T^{2} + 264 T^{3} + 2851 T^{4} + 264 p T^{5} + 78 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 12 T + 154 T^{2} - 1080 T^{3} + 7608 T^{4} - 1080 p T^{5} + 154 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 12 T + 172 T^{2} - 1224 T^{3} + 9747 T^{4} - 1224 p T^{5} + 172 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 44 T^{2} - 144 T^{3} + 1515 T^{4} - 144 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 232 T^{2} + 24668 T^{4} - 1646616 T^{6} + 80449574 T^{8} - 1646616 p^{2} T^{10} + 24668 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 228 T^{2} + 27716 T^{4} - 2177844 T^{6} + 121057578 T^{8} - 2177844 p^{2} T^{10} + 27716 p^{4} T^{12} - 228 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 240 T^{2} + 29870 T^{4} - 2491584 T^{6} + 152740371 T^{8} - 2491584 p^{2} T^{10} + 29870 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 336 T^{2} + 53804 T^{4} - 5435952 T^{6} + 380644038 T^{8} - 5435952 p^{2} T^{10} + 53804 p^{4} T^{12} - 336 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 232 T^{2} + 27620 T^{4} - 2298936 T^{6} + 153904550 T^{8} - 2298936 p^{2} T^{10} + 27620 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 - 4 T + 170 T^{2} - 648 T^{3} + 15680 T^{4} - 648 p T^{5} + 170 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 408 T^{2} + 79516 T^{4} - 9832104 T^{6} + 848678022 T^{8} - 9832104 p^{2} T^{10} + 79516 p^{4} T^{12} - 408 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( 1 - 228 T^{2} + 27316 T^{4} - 3032532 T^{6} + 285218922 T^{8} - 3032532 p^{2} T^{10} + 27316 p^{4} T^{12} - 228 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 8 T + 342 T^{2} - 1980 T^{3} + 42976 T^{4} - 1980 p T^{5} + 342 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 636 T^{2} + 2050 p T^{4} - 30885600 T^{6} + 3378509787 T^{8} - 30885600 p^{2} T^{10} + 2050 p^{5} T^{12} - 636 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 4 T + 242 T^{2} + 480 T^{3} + 29531 T^{4} + 480 p T^{5} + 242 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.40702343777284369962568748266, −4.35533344514634358102715769691, −4.15800471821447003527447527546, −3.74108699831619433969221763705, −3.70786599476122457337997235075, −3.62481453767660319133125706302, −3.45743263289984362994842730048, −3.38249334343157613925314100322, −3.33482438473675364380058580700, −2.86529054947943651120294135883, −2.80600523071883770270636340710, −2.73462550893097844758608523417, −2.56923248149098000941018535077, −2.54913443282969169773131765686, −2.49806716377531580815245287477, −2.21984203383152556334696108320, −2.16836567291346596553630627317, −1.82320531252015506017998101927, −1.57679613851361815560767927671, −1.24306642227288058787076826558, −1.15825688715175732741425867766, −0.977631359004136919407526889716, −0.917346323404996461291056374000, −0.873114187844279275780805203512, −0.35655123381243435198889122015,
0.35655123381243435198889122015, 0.873114187844279275780805203512, 0.917346323404996461291056374000, 0.977631359004136919407526889716, 1.15825688715175732741425867766, 1.24306642227288058787076826558, 1.57679613851361815560767927671, 1.82320531252015506017998101927, 2.16836567291346596553630627317, 2.21984203383152556334696108320, 2.49806716377531580815245287477, 2.54913443282969169773131765686, 2.56923248149098000941018535077, 2.73462550893097844758608523417, 2.80600523071883770270636340710, 2.86529054947943651120294135883, 3.33482438473675364380058580700, 3.38249334343157613925314100322, 3.45743263289984362994842730048, 3.62481453767660319133125706302, 3.70786599476122457337997235075, 3.74108699831619433969221763705, 4.15800471821447003527447527546, 4.35533344514634358102715769691, 4.40702343777284369962568748266
Plot not available for L-functions of degree greater than 10.
