| L(s) = 1 | + 2·4-s − 22·7-s − 58·13-s − 60·16-s + 58·19-s − 44·28-s − 536·31-s − 166·37-s + 464·43-s − 323·49-s − 116·52-s + 1.53e3·61-s − 248·64-s + 1.02e3·67-s − 274·73-s + 116·76-s − 950·79-s + 1.27e3·91-s − 1.64e3·97-s − 1.67e3·103-s + 436·109-s + 1.32e3·112-s − 2.37e3·121-s − 1.07e3·124-s + 127-s + 131-s − 1.27e3·133-s + ⋯ |
| L(s) = 1 | + 1/4·4-s − 1.18·7-s − 1.23·13-s − 0.937·16-s + 0.700·19-s − 0.296·28-s − 3.10·31-s − 0.737·37-s + 1.64·43-s − 0.941·49-s − 0.309·52-s + 3.21·61-s − 0.484·64-s + 1.86·67-s − 0.439·73-s + 0.175·76-s − 1.35·79-s + 1.46·91-s − 1.71·97-s − 1.60·103-s + 0.383·109-s + 1.11·112-s − 1.78·121-s − 0.776·124-s + 0.000698·127-s + 0.000666·131-s − 0.831·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 11 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2374 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 29 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7234 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 29 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 17134 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 24950 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 268 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 83 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 64114 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 232 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 55294 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 204442 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 327526 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 767 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 511 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 207790 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 137 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 475 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 810646 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 1345138 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 821 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.752970009954713032302948319623, −9.662703462558308981801551607456, −8.951491100615439913527481549114, −8.884288458983274553116788075008, −8.083844281435676251140585582200, −7.50583739846252517121368041519, −7.21997759836997549781354511058, −6.82067601573393648532902699085, −6.49657974974853502813010212677, −5.75816169886954256340852881003, −5.24975331191266595583226346327, −5.11098257936974426760877283297, −4.05363359074120629316676400727, −3.83921208981546362992786668310, −3.16119769787726033380519990669, −2.50407051501345297346275278927, −2.14251290417858218891827752121, −1.21665495723127506118951012085, 0, 0,
1.21665495723127506118951012085, 2.14251290417858218891827752121, 2.50407051501345297346275278927, 3.16119769787726033380519990669, 3.83921208981546362992786668310, 4.05363359074120629316676400727, 5.11098257936974426760877283297, 5.24975331191266595583226346327, 5.75816169886954256340852881003, 6.49657974974853502813010212677, 6.82067601573393648532902699085, 7.21997759836997549781354511058, 7.50583739846252517121368041519, 8.083844281435676251140585582200, 8.884288458983274553116788075008, 8.951491100615439913527481549114, 9.662703462558308981801551607456, 9.752970009954713032302948319623
