| L(s) = 1 | − 2·3-s − 6·4-s − 4·7-s + 4·9-s + 12·12-s + 8·13-s + 13·16-s − 16·19-s + 8·21-s − 4·27-s + 24·28-s + 8·31-s − 24·36-s − 28·37-s − 16·39-s + 36·43-s − 26·48-s + 8·49-s − 48·52-s + 32·57-s + 28·61-s − 16·63-s − 6·64-s − 40·67-s − 28·73-s + 96·76-s + 16·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3·4-s − 1.51·7-s + 4/3·9-s + 3.46·12-s + 2.21·13-s + 13/4·16-s − 3.67·19-s + 1.74·21-s − 0.769·27-s + 4.53·28-s + 1.43·31-s − 4·36-s − 4.60·37-s − 2.56·39-s + 5.48·43-s − 3.75·48-s + 8/7·49-s − 6.65·52-s + 4.23·57-s + 3.58·61-s − 2.01·63-s − 3/4·64-s − 4.88·67-s − 3.27·73-s + 11.0·76-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05507817592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05507817592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 2 T - 4 T^{3} - 5 T^{4} - 4 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 2 | \( ( 1 + 3 T^{2} + 7 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 - 22 T^{4} + 939 T^{8} - 22 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 2 T + 2 T^{2} - 24 T^{3} - 73 T^{4} - 24 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 24 T^{2} + 338 T^{4} + 3504 T^{6} + 29907 T^{8} + 3504 p^{2} T^{10} + 338 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 38 T^{2} + 613 T^{4} - 9614 T^{6} + 189724 T^{8} - 9614 p^{2} T^{10} + 613 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 8 T + 20 T^{2} - 60 T^{3} - 649 T^{4} - 60 p T^{5} + 20 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( 1 + 34 T^{2} - 707 T^{4} + 6154 T^{6} + 1791292 T^{8} + 6154 p^{2} T^{10} - 707 p^{4} T^{12} + 34 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 4 T + 8 T^{2} - 36 T^{3} - 322 T^{4} - 36 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 14 T + 113 T^{2} + 18 p T^{3} + 104 p T^{4} + 18 p^{2} T^{5} + 113 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 54 T^{2} + 221 T^{4} + 40554 T^{6} - 627828 T^{8} + 40554 p^{2} T^{10} + 221 p^{4} T^{12} - 54 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 18 T + 212 T^{2} - 1872 T^{3} + 13611 T^{4} - 1872 p T^{5} + 212 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 5500 T^{4} + 14557062 T^{8} - 5500 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 190 T^{2} + 14535 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 24 T^{2} - 3202 T^{4} - 81456 T^{6} + 70227 T^{8} - 81456 p^{2} T^{10} - 3202 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 20 T + 164 T^{2} + 564 T^{3} + 359 T^{4} + 564 p T^{5} + 164 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 - 24 T^{2} - 5554 T^{4} + 137904 T^{6} + 8572707 T^{8} + 137904 p^{2} T^{10} - 5554 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 14 T + 98 T^{2} + 1176 T^{3} + 13991 T^{4} + 1176 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 83 | \( 1 + 21212 T^{4} + 202731366 T^{8} + 21212 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( 1 - 24 T^{2} + 9026 T^{4} - 212016 T^{6} + 16818147 T^{8} - 212016 p^{2} T^{10} + 9026 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 10 T + 2 p T^{2} - 2124 T^{3} + 23279 T^{4} - 2124 p T^{5} + 2 p^{3} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−8.461159458642994795156395746183, −8.045450346075109607081572456416, −7.65224176197566244321321150969, −7.47799291026492365417143583494, −7.37041387682648272866778873398, −7.19487632050671802615762527029, −6.74412489325006960992443518691, −6.56108889588994508196361140845, −6.50642671967879362419518081301, −6.32027748736098314606872610873, −6.05142661502583991481500636523, −5.91572767614074564480228031070, −5.72918964101977330788315056569, −5.43884288613253686495927374509, −5.04852373394208917471091899464, −4.95599865204656729193849253605, −4.66855553541343769555966900722, −4.37483855784114194924410955852, −4.08471658587654049200097916742, −3.94754720668736229970447449288, −3.93268514198483980193294624806, −3.81856285014947763266534833046, −2.85080616848113840615003351245, −2.83605023072791147539139764242, −1.84816786409955800015425415144,
1.84816786409955800015425415144, 2.83605023072791147539139764242, 2.85080616848113840615003351245, 3.81856285014947763266534833046, 3.93268514198483980193294624806, 3.94754720668736229970447449288, 4.08471658587654049200097916742, 4.37483855784114194924410955852, 4.66855553541343769555966900722, 4.95599865204656729193849253605, 5.04852373394208917471091899464, 5.43884288613253686495927374509, 5.72918964101977330788315056569, 5.91572767614074564480228031070, 6.05142661502583991481500636523, 6.32027748736098314606872610873, 6.50642671967879362419518081301, 6.56108889588994508196361140845, 6.74412489325006960992443518691, 7.19487632050671802615762527029, 7.37041387682648272866778873398, 7.47799291026492365417143583494, 7.65224176197566244321321150969, 8.045450346075109607081572456416, 8.461159458642994795156395746183
Plot not available for L-functions of degree greater than 10.
