| L(s) = 1 | + 3·4-s − 6·9-s − 4·11-s + 4·16-s + 16·23-s − 10·25-s − 18·36-s − 18·37-s − 12·44-s + 7·49-s + 30·53-s + 8·67-s − 48·71-s − 40·79-s + 9·81-s + 48·92-s + 24·99-s − 30·100-s + 100·107-s − 4·113-s + 11·121-s + 127-s + 131-s + 137-s + 139-s − 24·144-s − 54·148-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 2·9-s − 1.20·11-s + 16-s + 3.33·23-s − 2·25-s − 3·36-s − 2.95·37-s − 1.80·44-s + 49-s + 4.12·53-s + 0.977·67-s − 5.69·71-s − 4.50·79-s + 81-s + 5.00·92-s + 2.41·99-s − 3·100-s + 9.66·107-s − 0.376·113-s + 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2·144-s − 4.43·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6531912872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6531912872\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | \( 1 + 4 T + 5 T^{2} - 24 T^{3} - 151 T^{4} - 24 p T^{5} + 5 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 2 | \( ( 1 - T - T^{2} + 3 T^{3} - T^{4} + 3 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )( 1 + T - T^{2} - 3 T^{3} - T^{4} - 3 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} ) \) |
| 3 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{2}( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 8 T + 41 T^{2} - 144 T^{3} + 209 T^{4} - 144 p T^{5} + 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 2 T - 25 T^{2} + 108 T^{3} + 509 T^{4} + 108 p T^{5} - 25 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )( 1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} - 108 p T^{5} - 25 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 31 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 6 T + p T^{2} )^{4}( 1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} + 228 p T^{5} - p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 41 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 12 T + 101 T^{2} - 696 T^{3} + 4009 T^{4} - 696 p T^{5} + 101 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )( 1 + 12 T + 101 T^{2} + 696 T^{3} + 4009 T^{4} + 696 p T^{5} + 101 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 47 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + 47 T^{2} - 60 T^{3} - 3091 T^{4} - 60 p T^{5} + 47 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 59 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 4 T - 51 T^{2} + 472 T^{3} + 1529 T^{4} + 472 p T^{5} - 51 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 16 T + p T^{2} )^{4}( 1 - 16 T + 185 T^{2} - 1824 T^{3} + 16049 T^{4} - 1824 p T^{5} + 185 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 73 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 8 T + p T^{2} )^{4}( 1 + 8 T - 15 T^{2} - 752 T^{3} - 4831 T^{4} - 752 p T^{5} - 15 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 83 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−7.11620769384632856283985762010, −6.81621386434999215812144164943, −6.71377062309938648770693824469, −6.26079259088659468758450560814, −6.22382220924576802860251920454, −6.02238261084894950157651000341, −5.80939066583853134185975395579, −5.66869429989558825096445566019, −5.57137144896879636119229479647, −5.46482451547966621051434341265, −5.23099003626911556878673020908, −4.91626339372464437211532315016, −4.78608684352023633398155941664, −4.48764994586875992624151151943, −4.39772082706543120086919542313, −3.92126450689799338293445164155, −3.68223452809524052525377654786, −3.44143414793870589705821960940, −3.20489147426626679176319633097, −2.96548008747228776688903917910, −2.78303236013012806077080009213, −2.58569582382856627395194458269, −2.23595303929477303313458774310, −1.94007870270729304110028665313, −1.39278900581271508679553308498,
1.39278900581271508679553308498, 1.94007870270729304110028665313, 2.23595303929477303313458774310, 2.58569582382856627395194458269, 2.78303236013012806077080009213, 2.96548008747228776688903917910, 3.20489147426626679176319633097, 3.44143414793870589705821960940, 3.68223452809524052525377654786, 3.92126450689799338293445164155, 4.39772082706543120086919542313, 4.48764994586875992624151151943, 4.78608684352023633398155941664, 4.91626339372464437211532315016, 5.23099003626911556878673020908, 5.46482451547966621051434341265, 5.57137144896879636119229479647, 5.66869429989558825096445566019, 5.80939066583853134185975395579, 6.02238261084894950157651000341, 6.22382220924576802860251920454, 6.26079259088659468758450560814, 6.71377062309938648770693824469, 6.81621386434999215812144164943, 7.11620769384632856283985762010
Plot not available for L-functions of degree greater than 10.
