L(s) = 1 | − 4·2-s + 6·4-s + 4·8-s − 10·9-s + 4·11-s − 35·16-s + 40·18-s − 16·22-s − 12·23-s − 26·25-s − 8·29-s + 60·32-s − 60·36-s + 8·37-s − 32·43-s + 24·44-s + 48·46-s + 104·50-s − 4·53-s + 32·58-s − 14·64-s − 20·67-s − 8·71-s − 40·72-s − 32·74-s − 4·79-s + 33·81-s + ⋯ |
L(s) = 1 | − 2.82·2-s + 3·4-s + 1.41·8-s − 3.33·9-s + 1.20·11-s − 8.75·16-s + 9.42·18-s − 3.41·22-s − 2.50·23-s − 5.19·25-s − 1.48·29-s + 10.6·32-s − 10·36-s + 1.31·37-s − 4.87·43-s + 3.61·44-s + 7.07·46-s + 14.7·50-s − 0.549·53-s + 4.20·58-s − 7/4·64-s − 2.44·67-s − 0.949·71-s − 4.71·72-s − 3.71·74-s − 0.450·79-s + 11/3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( ( 1 + p T + 3 T^{2} - T^{4} + 3 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} )^{2} \) |
| 3 | \( 1 + 10 T^{2} + 67 T^{4} + 34 p^{2} T^{6} + 1057 T^{8} + 34 p^{4} T^{10} + 67 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 + 26 T^{2} + 311 T^{4} + 2358 T^{6} + 13281 T^{8} + 2358 p^{2} T^{10} + 311 p^{4} T^{12} + 26 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 2 T + 39 T^{2} - 54 T^{3} + 611 T^{4} - 54 p T^{5} + 39 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 78 T^{2} + 3142 T^{4} + 85520 T^{6} + 1694711 T^{8} + 85520 p^{2} T^{10} + 3142 p^{4} T^{12} + 78 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 + 58 T^{2} + 1703 T^{4} + 44478 T^{6} + 996825 T^{8} + 44478 p^{2} T^{10} + 1703 p^{4} T^{12} + 58 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 6 T + 78 T^{2} + 358 T^{3} + 2630 T^{4} + 358 p T^{5} + 78 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 4 T + 31 T^{2} + 158 T^{3} + 1897 T^{4} + 158 p T^{5} + 31 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 178 T^{2} + 15406 T^{4} + 835752 T^{6} + 30991415 T^{8} + 835752 p^{2} T^{10} + 15406 p^{4} T^{12} + 178 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 4 T + 62 T^{2} - 48 T^{3} + 1470 T^{4} - 48 p T^{5} + 62 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 + 152 T^{2} + 6780 T^{4} - 92120 T^{6} - 15043642 T^{8} - 92120 p^{2} T^{10} + 6780 p^{4} T^{12} + 152 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 16 T + 157 T^{2} + 1020 T^{3} + 6599 T^{4} + 1020 p T^{5} + 157 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 150 T^{2} + 11494 T^{4} + 707936 T^{6} + 37180919 T^{8} + 707936 p^{2} T^{10} + 11494 p^{4} T^{12} + 150 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 2 T + 132 T^{2} + 248 T^{3} + 8645 T^{4} + 248 p T^{5} + 132 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 278 T^{2} + 36286 T^{4} + 3136752 T^{6} + 207469775 T^{8} + 3136752 p^{2} T^{10} + 36286 p^{4} T^{12} + 278 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 + 380 T^{2} + 67856 T^{4} + 7443748 T^{6} + 547986382 T^{8} + 7443748 p^{2} T^{10} + 67856 p^{4} T^{12} + 380 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 10 T + 262 T^{2} + 1788 T^{3} + 25847 T^{4} + 1788 p T^{5} + 262 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 4 T + 118 T^{2} + 584 T^{3} + 12630 T^{4} + 584 p T^{5} + 118 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 156 T^{2} + 7232 T^{4} + 2020 p T^{6} + 11253390 T^{8} + 2020 p^{3} T^{10} + 7232 p^{4} T^{12} + 156 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 2 T + 38 T^{2} - 238 T^{3} + 1686 T^{4} - 238 p T^{5} + 38 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 314 T^{2} + 46430 T^{4} + 4837464 T^{6} + 427877847 T^{8} + 4837464 p^{2} T^{10} + 46430 p^{4} T^{12} + 314 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 + 106 T^{2} + 17871 T^{4} + 2077238 T^{6} + 155830553 T^{8} + 2077238 p^{2} T^{10} + 17871 p^{4} T^{12} + 106 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 + 412 T^{2} + 82235 T^{4} + 11265530 T^{6} + 1213308829 T^{8} + 11265530 p^{2} T^{10} + 82235 p^{4} T^{12} + 412 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.65311425195365555564267740337, −3.31543735620151600451248164613, −3.22275343116121289590932936374, −3.17470227187697450442090285092, −3.09905889221511071623218983442, −3.06056664899500846314100733943, −2.99914298368258158543816200282, −2.84845562450484453325423784750, −2.66272271148759598725959920376, −2.40045960394371959364580152623, −2.29346709031859273786384407873, −2.26716707430024564372116403759, −2.16246752827108683333604561905, −2.08120445065263952018945724395, −2.07759574020696943631755688084, −1.92849402637800627637645726706, −1.67159926235190607789738035251, −1.60354746429304010226338129061, −1.58149951382022750043497983530, −1.54893250462767763179822117152, −1.24153782282928861169600609983, −1.17678426297016803186236697033, −1.00658413089117427348390392908, −0.977387256770598168798957325041, −0.816178458964244307761123419204, 0, 0, 0, 0, 0, 0, 0, 0,
0.816178458964244307761123419204, 0.977387256770598168798957325041, 1.00658413089117427348390392908, 1.17678426297016803186236697033, 1.24153782282928861169600609983, 1.54893250462767763179822117152, 1.58149951382022750043497983530, 1.60354746429304010226338129061, 1.67159926235190607789738035251, 1.92849402637800627637645726706, 2.07759574020696943631755688084, 2.08120445065263952018945724395, 2.16246752827108683333604561905, 2.26716707430024564372116403759, 2.29346709031859273786384407873, 2.40045960394371959364580152623, 2.66272271148759598725959920376, 2.84845562450484453325423784750, 2.99914298368258158543816200282, 3.06056664899500846314100733943, 3.09905889221511071623218983442, 3.17470227187697450442090285092, 3.22275343116121289590932936374, 3.31543735620151600451248164613, 3.65311425195365555564267740337
Plot not available for L-functions of degree greater than 10.