| L(s) = 1 | + 8·2-s + 20·4-s − 24·8-s + 8·9-s − 4·11-s − 246·16-s + 64·18-s − 32·22-s − 24·23-s − 4·25-s + 16·29-s − 408·32-s + 160·36-s − 16·37-s − 12·43-s − 80·44-s − 192·46-s − 32·50-s + 12·53-s + 128·58-s + 756·64-s − 4·67-s − 24·71-s − 192·72-s − 128·74-s + 28·79-s + 34·81-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 10·4-s − 8.48·8-s + 8/3·9-s − 1.20·11-s − 61.5·16-s + 15.0·18-s − 6.82·22-s − 5.00·23-s − 4/5·25-s + 2.97·29-s − 72.1·32-s + 80/3·36-s − 2.63·37-s − 1.82·43-s − 12.0·44-s − 28.3·46-s − 4.52·50-s + 1.64·53-s + 16.8·58-s + 94.5·64-s − 0.488·67-s − 2.84·71-s − 22.6·72-s − 14.8·74-s + 3.15·79-s + 34/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.643226037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.643226037\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + 20 T^{2} + 231 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 2 | \( ( 1 - T + p T^{2} )^{8} \) |
| 3 | \( ( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 + 4 T^{2} - 3 p T^{4} - 76 T^{6} - 64 T^{8} - 76 p^{2} T^{10} - 3 p^{5} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 2 T + 4 T^{2} - 4 p T^{3} - 15 p T^{4} - 4 p^{2} T^{5} + 4 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 36 T^{2} + 695 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 8 T + 13 T^{2} + 56 T^{3} - 96 T^{4} + 56 p T^{5} + 13 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 60 T^{2} + 2639 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 55 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( 1 - 60 T^{2} + 1201 T^{4} + 57780 T^{6} - 3122160 T^{8} + 57780 p^{2} T^{10} + 1201 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 6 T - 36 T^{2} - 84 T^{3} + 1787 T^{4} - 84 p T^{5} - 36 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 6 T + 13 T^{2} + 498 T^{3} - 4188 T^{4} + 498 p T^{5} + 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 80 T^{2} + 2674 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( 1 - 172 T^{2} + 15873 T^{4} - 1078268 T^{6} + 64063616 T^{8} - 1078268 p^{2} T^{10} + 15873 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 2 T - 108 T^{2} - 44 T^{3} + 7787 T^{4} - 44 p T^{5} - 108 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 100 T^{2} + 729 T^{4} + 1900 p T^{6} + 800 p^{2} T^{8} + 1900 p^{3} T^{10} + 729 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 14 T + 12 T^{2} - 364 T^{3} + 12131 T^{4} - 364 p T^{5} + 12 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 16 T^{2} + 14434 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 176 T^{2} + 21567 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.51043433773061834841750889285, −4.48161847172869964329144989746, −4.39011797812131996683950273538, −4.37222800816157669300173479210, −3.98377759264676169751721074122, −3.97754157432962080753699380487, −3.97650937219504191883003559191, −3.90354171036263999738015430229, −3.70944062101175525304037601134, −3.47666847062815129371261462067, −3.47495647249182616856399331900, −3.46156387207036421448766120085, −3.07111303589758090911808126105, −2.91023719803409712959394488642, −2.89706855548721225433193763047, −2.81479634220447093534516221044, −2.31825306463554068587329085815, −2.27731004662655525782284625794, −2.26018287278257359617212917489, −1.65642452258053526831642049145, −1.63437641373743660032483214025, −1.06033177312534432396940287417, −0.789510091918761586153806048591, −0.34712572860200048137294165223, −0.18803524849220343707046100301,
0.18803524849220343707046100301, 0.34712572860200048137294165223, 0.789510091918761586153806048591, 1.06033177312534432396940287417, 1.63437641373743660032483214025, 1.65642452258053526831642049145, 2.26018287278257359617212917489, 2.27731004662655525782284625794, 2.31825306463554068587329085815, 2.81479634220447093534516221044, 2.89706855548721225433193763047, 2.91023719803409712959394488642, 3.07111303589758090911808126105, 3.46156387207036421448766120085, 3.47495647249182616856399331900, 3.47666847062815129371261462067, 3.70944062101175525304037601134, 3.90354171036263999738015430229, 3.97650937219504191883003559191, 3.97754157432962080753699380487, 3.98377759264676169751721074122, 4.37222800816157669300173479210, 4.39011797812131996683950273538, 4.48161847172869964329144989746, 4.51043433773061834841750889285
Plot not available for L-functions of degree greater than 10.
