| L(s) = 1 | + 2·3-s + 3·4-s − 3·9-s + 6·12-s + 9·16-s + 8·17-s + 4·23-s + 15·25-s − 28·27-s + 2·29-s − 9·36-s − 6·43-s + 18·48-s − 4·49-s + 16·51-s + 4·53-s + 16·61-s + 26·64-s + 24·68-s + 8·69-s + 30·75-s − 52·79-s − 34·81-s + 4·87-s + 12·92-s + 45·100-s + 18·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 3/2·4-s − 9-s + 1.73·12-s + 9/4·16-s + 1.94·17-s + 0.834·23-s + 3·25-s − 5.38·27-s + 0.371·29-s − 3/2·36-s − 0.914·43-s + 2.59·48-s − 4/7·49-s + 2.24·51-s + 0.549·53-s + 2.04·61-s + 13/4·64-s + 2.91·68-s + 0.963·69-s + 3.46·75-s − 5.85·79-s − 3.77·81-s + 0.428·87-s + 1.25·92-s + 9/2·100-s + 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.44996530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.44996530\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( ( 1 + T^{2} )^{4} \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3 T^{2} + T^{6} + 13 T^{8} + p^{2} T^{10} - 3 p^{6} T^{14} + p^{8} T^{16} \) |
| 3 | \( ( 1 - T + p T^{2} + 7 T^{3} - 2 p T^{4} + 7 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 - 3 p T^{2} + 18 p T^{4} - 208 T^{6} + 161 T^{8} - 208 p^{2} T^{10} + 18 p^{5} T^{12} - 3 p^{7} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 69 T^{2} + 2235 T^{4} - 44347 T^{6} + 589196 T^{8} - 44347 p^{2} T^{10} + 2235 p^{4} T^{12} - 69 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 4 T + 56 T^{2} - 144 T^{3} + 1273 T^{4} - 144 p T^{5} + 56 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 41 T^{2} + 1479 T^{4} - 33939 T^{6} + 773640 T^{8} - 33939 p^{2} T^{10} + 1479 p^{4} T^{12} - 41 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 2 T + 54 T^{2} - 82 T^{3} + 1498 T^{4} - 82 p T^{5} + 54 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - T + 94 T^{2} - 108 T^{3} + 3765 T^{4} - 108 p T^{5} + 94 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 206 T^{2} + 19581 T^{4} - 1120926 T^{6} + 42292496 T^{8} - 1120926 p^{2} T^{10} + 19581 p^{4} T^{12} - 206 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 - 230 T^{2} + 24817 T^{4} - 1647054 T^{6} + 73485364 T^{8} - 1647054 p^{2} T^{10} + 24817 p^{4} T^{12} - 230 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 - 142 T^{2} + 12441 T^{4} - 775594 T^{6} + 36012204 T^{8} - 775594 p^{2} T^{10} + 12441 p^{4} T^{12} - 142 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 3 T + 169 T^{2} + 379 T^{3} + 10834 T^{4} + 379 p T^{5} + 169 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 270 T^{2} + 34713 T^{4} - 2808998 T^{6} + 156998836 T^{8} - 2808998 p^{2} T^{10} + 34713 p^{4} T^{12} - 270 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 2 T + 72 T^{2} + 572 T^{3} + 625 T^{4} + 572 p T^{5} + 72 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 254 T^{2} + 33613 T^{4} - 3067390 T^{6} + 208534560 T^{8} - 3067390 p^{2} T^{10} + 33613 p^{4} T^{12} - 254 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 8 T + 229 T^{2} - 1288 T^{3} + 20396 T^{4} - 1288 p T^{5} + 229 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 78 T^{2} + 15725 T^{4} - 827054 T^{6} + 97483072 T^{8} - 827054 p^{2} T^{10} + 15725 p^{4} T^{12} - 78 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 - 340 T^{2} + 60148 T^{4} - 6941228 T^{6} + 575927382 T^{8} - 6941228 p^{2} T^{10} + 60148 p^{4} T^{12} - 340 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 282 T^{2} + 36049 T^{4} - 2796098 T^{6} + 189769156 T^{8} - 2796098 p^{2} T^{10} + 36049 p^{4} T^{12} - 282 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 26 T + 450 T^{2} + 5138 T^{3} + 50938 T^{4} + 5138 p T^{5} + 450 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 470 T^{2} + 104837 T^{4} - 14701286 T^{6} + 1438713904 T^{8} - 14701286 p^{2} T^{10} + 104837 p^{4} T^{12} - 470 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 569 T^{2} + 150619 T^{4} - 24317907 T^{6} + 2619263152 T^{8} - 24317907 p^{2} T^{10} + 150619 p^{4} T^{12} - 569 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 157 T^{2} + 31611 T^{4} - 3554575 T^{6} + 450491376 T^{8} - 3554575 p^{2} T^{10} + 31611 p^{4} T^{12} - 157 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.04588260064449802515949667203, −3.90088120398658949254749553004, −3.88349549384318340142674053311, −3.86856630570827302328373768268, −3.63482554156963940977545663370, −3.33189292157050453225091944178, −3.32410106011350187701140812195, −3.15641269659373116558184295480, −3.04262603085237798912218088604, −2.94126698529732879537382977210, −2.86690805806240668517075588308, −2.80031010405854466933858165534, −2.52610896471531388508723719255, −2.46557437504673626176629000262, −2.43550793323338630804984011593, −2.10858386739300633156347135544, −1.79193834449114359712912237937, −1.77481783129762205720600333027, −1.69447839823033993183584966495, −1.34879991423550882966259992644, −1.34833379215562490137303545141, −1.15174057452901347162689897001, −0.68851975095444636229562804839, −0.66656359102685495761752157929, −0.26772572594525316824304681138,
0.26772572594525316824304681138, 0.66656359102685495761752157929, 0.68851975095444636229562804839, 1.15174057452901347162689897001, 1.34833379215562490137303545141, 1.34879991423550882966259992644, 1.69447839823033993183584966495, 1.77481783129762205720600333027, 1.79193834449114359712912237937, 2.10858386739300633156347135544, 2.43550793323338630804984011593, 2.46557437504673626176629000262, 2.52610896471531388508723719255, 2.80031010405854466933858165534, 2.86690805806240668517075588308, 2.94126698529732879537382977210, 3.04262603085237798912218088604, 3.15641269659373116558184295480, 3.32410106011350187701140812195, 3.33189292157050453225091944178, 3.63482554156963940977545663370, 3.86856630570827302328373768268, 3.88349549384318340142674053311, 3.90088120398658949254749553004, 4.04588260064449802515949667203
Plot not available for L-functions of degree greater than 10.
