| L(s) = 1 | + 4-s + 9-s + 2·11-s + 2·19-s + 32·29-s − 12·31-s + 36-s + 36·41-s + 2·44-s − 2·49-s + 24·59-s + 8·61-s − 64-s − 56·71-s + 2·76-s + 8·79-s − 4·89-s + 2·99-s − 20·109-s + 32·116-s + 23·121-s − 12·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s + 0.603·11-s + 0.458·19-s + 5.94·29-s − 2.15·31-s + 1/6·36-s + 5.62·41-s + 0.301·44-s − 2/7·49-s + 3.12·59-s + 1.02·61-s − 1/8·64-s − 6.64·71-s + 0.229·76-s + 0.900·79-s − 0.423·89-s + 0.201·99-s − 1.91·109-s + 2.97·116-s + 2.09·121-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.829982704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.829982704\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 85 T^{2} + 5016 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 105 T^{2} + 8216 T^{4} + 105 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 10 T^{2} - 4389 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 50 T^{2} - 2829 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.11672140714262161287270767643, −6.86515999285472634002441734167, −6.86151403788306634063673009953, −6.33727116123001502217389682669, −6.12884177882849378549241655912, −6.08544207288135951721192789406, −5.71567551808863462555738920185, −5.67613177828573388864680577364, −5.45504625595180649407623300544, −4.82261142983218330957562986479, −4.65167387274617843381542071030, −4.57221651081651535239326906445, −4.45978841307789148612285048833, −4.03340493984169817848362607907, −3.91675974534516295978698803985, −3.37676575614118400055559400822, −3.32080898977328157369780591679, −2.77062533509549211143650263490, −2.63763530926095147164908056660, −2.42725576384748579702659078251, −2.32106329679117397535970729010, −1.44921933712682174067501014468, −1.35123098113178104972523446365, −0.923491946537888398229741586062, −0.64353512510729536513562094248,
0.64353512510729536513562094248, 0.923491946537888398229741586062, 1.35123098113178104972523446365, 1.44921933712682174067501014468, 2.32106329679117397535970729010, 2.42725576384748579702659078251, 2.63763530926095147164908056660, 2.77062533509549211143650263490, 3.32080898977328157369780591679, 3.37676575614118400055559400822, 3.91675974534516295978698803985, 4.03340493984169817848362607907, 4.45978841307789148612285048833, 4.57221651081651535239326906445, 4.65167387274617843381542071030, 4.82261142983218330957562986479, 5.45504625595180649407623300544, 5.67613177828573388864680577364, 5.71567551808863462555738920185, 6.08544207288135951721192789406, 6.12884177882849378549241655912, 6.33727116123001502217389682669, 6.86151403788306634063673009953, 6.86515999285472634002441734167, 7.11672140714262161287270767643
