| L(s) = 1 | − 2·3-s + 6·4-s + 4·7-s + 4·9-s − 12·12-s + 13·16-s − 8·19-s − 8·21-s − 4·27-s + 24·28-s − 8·31-s + 24·36-s − 32·37-s − 36·43-s − 26·48-s + 8·49-s + 16·57-s + 28·61-s + 16·63-s + 6·64-s − 32·67-s + 28·73-s − 48·76-s + 16·79-s + 5·81-s − 48·84-s + 16·93-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3·4-s + 1.51·7-s + 4/3·9-s − 3.46·12-s + 13/4·16-s − 1.83·19-s − 1.74·21-s − 0.769·27-s + 4.53·28-s − 1.43·31-s + 4·36-s − 5.26·37-s − 5.48·43-s − 3.75·48-s + 8/7·49-s + 2.11·57-s + 3.58·61-s + 2.01·63-s + 3/4·64-s − 3.90·67-s + 3.27·73-s − 5.50·76-s + 1.80·79-s + 5/9·81-s − 5.23·84-s + 1.65·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.461332249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.461332249\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 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 | \( ( 1 + 4 T + 20 T^{2} + 108 T^{3} + 263 T^{4} + 108 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 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 | \( ( 1 + 16 T + 113 T^{2} + 12 p T^{3} + 44 p T^{4} + 12 p^{2} T^{5} + 113 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 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 | \( ( 1 + 16 T + 164 T^{2} + 1308 T^{3} + 10007 T^{4} + 1308 p T^{5} + 164 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 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 | \( ( 1 - 26 T + 2 p T^{2} + 1332 T^{3} - 32593 T^{4} + 1332 p T^{5} + 2 p^{3} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.86405098504529935513702706169, −4.83263945375793450183228143678, −4.67282291465948754249829862707, −4.62458216902202625421554139472, −4.09782990756930129654610203537, −3.93150212020858412277039653489, −3.92077542360034668922096108526, −3.76735155073012539095355093411, −3.59245078878281311721689405111, −3.51029497278680775378630322994, −3.43704573978074693373045962452, −3.23103242293428986240220312769, −2.99148776621567670188964490390, −2.61369643401669024792439966885, −2.57997850292195011871378432332, −2.32710235232245273061248473938, −2.23506878727769997502493573503, −1.95431385992676353423379905981, −1.95421857043950002391161580034, −1.78922642333466199372968203516, −1.51203502726482736753009556803, −1.45933613688877365227724116931, −1.43618569632707664073550371686, −0.46761797611954147487836630463, −0.44291226471568165609263552419,
0.44291226471568165609263552419, 0.46761797611954147487836630463, 1.43618569632707664073550371686, 1.45933613688877365227724116931, 1.51203502726482736753009556803, 1.78922642333466199372968203516, 1.95421857043950002391161580034, 1.95431385992676353423379905981, 2.23506878727769997502493573503, 2.32710235232245273061248473938, 2.57997850292195011871378432332, 2.61369643401669024792439966885, 2.99148776621567670188964490390, 3.23103242293428986240220312769, 3.43704573978074693373045962452, 3.51029497278680775378630322994, 3.59245078878281311721689405111, 3.76735155073012539095355093411, 3.92077542360034668922096108526, 3.93150212020858412277039653489, 4.09782990756930129654610203537, 4.62458216902202625421554139472, 4.67282291465948754249829862707, 4.83263945375793450183228143678, 4.86405098504529935513702706169
Plot not available for L-functions of degree greater than 10.
