| L(s) = 1 | − 8·9-s − 32·13-s + 32·25-s − 16·37-s − 4·49-s − 16·73-s + 30·81-s + 16·97-s + 16·109-s + 256·117-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 512·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 8/3·9-s − 8.87·13-s + 32/5·25-s − 2.63·37-s − 4/7·49-s − 1.87·73-s + 10/3·81-s + 1.62·97-s + 1.53·109-s + 23.6·117-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 39.3·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5209106588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5209106588\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 12 T^{2} - 26 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 24 T^{2} + 226 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 60 T^{2} + 1942 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 60 T^{2} + 1702 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 84 T^{2} + 4822 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 108 T^{2} + 5974 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 180 T^{2} + 15622 T^{4} + 180 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 168 T^{2} + 20194 T^{4} + 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 252 T^{2} + 29158 T^{4} - 252 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−4.68774349523436148611786533459, −4.60358317178487391461030322792, −4.52151204689432941265831724335, −4.17876008620156716390738872404, −4.12690368845679156928238890074, −4.10242308405640643411985180278, −3.47155215969148118065310607293, −3.46065635417491917337146806182, −3.24175488480860312762508301777, −3.17653944485991332451710318583, −3.09634837561427023532095664143, −2.97557803021906792295090887207, −2.72714324522088687159536663149, −2.61179994072397058484919306686, −2.51725997783230688746864151443, −2.41562304396615535181400716163, −2.40592926154921393596040310944, −2.29858261603519240425404158228, −1.78823754687856873557089327133, −1.63114332403665162694278117867, −1.62982171533363401867850436269, −1.00223227001158987024373141102, −0.53611211424085824405630853413, −0.45951354822095963822983817039, −0.23209573272376829836668012719,
0.23209573272376829836668012719, 0.45951354822095963822983817039, 0.53611211424085824405630853413, 1.00223227001158987024373141102, 1.62982171533363401867850436269, 1.63114332403665162694278117867, 1.78823754687856873557089327133, 2.29858261603519240425404158228, 2.40592926154921393596040310944, 2.41562304396615535181400716163, 2.51725997783230688746864151443, 2.61179994072397058484919306686, 2.72714324522088687159536663149, 2.97557803021906792295090887207, 3.09634837561427023532095664143, 3.17653944485991332451710318583, 3.24175488480860312762508301777, 3.46065635417491917337146806182, 3.47155215969148118065310607293, 4.10242308405640643411985180278, 4.12690368845679156928238890074, 4.17876008620156716390738872404, 4.52151204689432941265831724335, 4.60358317178487391461030322792, 4.68774349523436148611786533459
Plot not available for L-functions of degree greater than 10.
