| L(s) = 1 | − 4·2-s + 8·4-s − 8·5-s − 16·8-s + 32·10-s + 36·16-s + 16·19-s − 64·20-s + 16·23-s + 28·25-s − 24·29-s − 64·32-s − 64·38-s + 128·40-s + 8·43-s − 64·46-s − 8·47-s − 4·49-s − 112·50-s + 96·58-s + 96·64-s + 8·67-s + 32·71-s + 8·73-s + 128·76-s − 288·80-s − 32·86-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 3.57·5-s − 5.65·8-s + 10.1·10-s + 9·16-s + 3.67·19-s − 14.3·20-s + 3.33·23-s + 28/5·25-s − 4.45·29-s − 11.3·32-s − 10.3·38-s + 20.2·40-s + 1.21·43-s − 9.43·46-s − 1.16·47-s − 4/7·49-s − 15.8·50-s + 12.6·58-s + 12·64-s + 0.977·67-s + 3.79·71-s + 0.936·73-s + 14.6·76-s − 32.1·80-s − 3.45·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1920786615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1920786615\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( ( 1 + 4 T + 2 p T^{2} + 32 T^{3} + 87 T^{4} + 32 p T^{5} + 2 p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 4 p T^{2} + 994 T^{4} - 15968 T^{6} + 198763 T^{8} - 15968 p^{2} T^{10} + 994 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 8 T^{2} + 228 T^{4} - 4504 T^{6} + 20006 T^{8} - 4504 p^{2} T^{10} + 228 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 19 | \( ( 1 - 8 T + 30 T^{2} - 112 T^{3} + 611 T^{4} - 112 p T^{5} + 30 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 8 T + 2 p T^{2} - 88 T^{3} + 231 T^{4} - 88 p T^{5} + 2 p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( 1 - 156 T^{2} + 12106 T^{4} - 610416 T^{6} + 22035219 T^{8} - 610416 p^{2} T^{10} + 12106 p^{4} T^{12} - 156 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 - 164 T^{2} + 14154 T^{4} - 829168 T^{6} + 35617331 T^{8} - 829168 p^{2} T^{10} + 14154 p^{4} T^{12} - 164 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 - 116 T^{2} + 11362 T^{4} - 660608 T^{6} + 32960203 T^{8} - 660608 p^{2} T^{10} + 11362 p^{4} T^{12} - 116 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 4 T + 84 T^{2} - 44 T^{3} + 3002 T^{4} - 44 p T^{5} + 84 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 4 T + 76 T^{2} + 44 T^{3} + 2154 T^{4} + 44 p T^{5} + 76 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 124 T^{2} - 192 T^{3} + 7734 T^{4} - 192 p T^{5} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 232 T^{2} + 28836 T^{4} - 2582456 T^{6} + 175948646 T^{8} - 2582456 p^{2} T^{10} + 28836 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 92 T^{2} + 6486 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 4 T + 228 T^{2} - 620 T^{3} + 21530 T^{4} - 620 p T^{5} + 228 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 16 T + 310 T^{2} - 3320 T^{3} + 33807 T^{4} - 3320 p T^{5} + 310 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 4 T + 156 T^{2} - 20 p T^{3} + 11402 T^{4} - 20 p^{2} T^{5} + 156 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 248 T^{2} + 37188 T^{4} - 3949096 T^{6} + 344639558 T^{8} - 3949096 p^{2} T^{10} + 37188 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 232 T^{2} + 34596 T^{4} - 3949112 T^{6} + 355979174 T^{8} - 3949112 p^{2} T^{10} + 34596 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 292 T^{2} + 47890 T^{4} - 6052768 T^{6} + 614802619 T^{8} - 6052768 p^{2} T^{10} + 47890 p^{4} T^{12} - 292 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 16 T + 300 T^{2} + 4208 T^{3} + 41318 T^{4} + 4208 p T^{5} + 300 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.89168634955581277742670355533, −3.75434347637364297473757051697, −3.70501310319092415716534079714, −3.56731113437477808217452593823, −3.54617555576433031783745282575, −3.53352953118510871126761390099, −3.09460878078627720020357216256, −3.02485049713733545723563031446, −2.99698224409194797630532426197, −2.94874870464311039219036779392, −2.81647653705923047026546317141, −2.68423024587464121861860401845, −2.36446085557178728466188200605, −2.20554456209787658899865059558, −2.19963682463442421976025197157, −1.67286391474213882435923073291, −1.63977919637158342466306862780, −1.63494795866894550576138989383, −1.22403549355326587184619081755, −1.10433200883185721856661276180, −0.959285290487153004088968285354, −0.71445347436695754353995810951, −0.52702002844814035940753440622, −0.43410459779587824454788793448, −0.18652625230188282050883066096,
0.18652625230188282050883066096, 0.43410459779587824454788793448, 0.52702002844814035940753440622, 0.71445347436695754353995810951, 0.959285290487153004088968285354, 1.10433200883185721856661276180, 1.22403549355326587184619081755, 1.63494795866894550576138989383, 1.63977919637158342466306862780, 1.67286391474213882435923073291, 2.19963682463442421976025197157, 2.20554456209787658899865059558, 2.36446085557178728466188200605, 2.68423024587464121861860401845, 2.81647653705923047026546317141, 2.94874870464311039219036779392, 2.99698224409194797630532426197, 3.02485049713733545723563031446, 3.09460878078627720020357216256, 3.53352953118510871126761390099, 3.54617555576433031783745282575, 3.56731113437477808217452593823, 3.70501310319092415716534079714, 3.75434347637364297473757051697, 3.89168634955581277742670355533
Plot not available for L-functions of degree greater than 10.
