| L(s) = 1 | + 8·5-s − 2·7-s − 28·11-s − 46·17-s + 20·19-s + 2·23-s − 2·25-s − 16·35-s + 136·43-s − 52·47-s − 95·49-s − 224·55-s − 80·61-s − 14·73-s + 56·77-s − 64·83-s − 368·85-s + 160·95-s − 28·101-s + 16·115-s + 92·119-s + 346·121-s − 344·125-s + 127-s + 131-s − 40·133-s + 137-s + ⋯ |
| L(s) = 1 | + 8/5·5-s − 2/7·7-s − 2.54·11-s − 2.70·17-s + 1.05·19-s + 2/23·23-s − 0.0799·25-s − 0.457·35-s + 3.16·43-s − 1.10·47-s − 1.93·49-s − 4.07·55-s − 1.31·61-s − 0.191·73-s + 8/11·77-s − 0.771·83-s − 4.32·85-s + 1.68·95-s − 0.277·101-s + 0.139·115-s + 0.773·119-s + 2.85·121-s − 2.75·125-s + 0.00787·127-s + 0.00763·131-s − 0.300·133-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.114577951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.114577951\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 20 T + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 77 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 23 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 23 p T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 878 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 1694 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2318 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 68 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 907 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6701 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8717 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9038 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 3086 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 862 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 9422 T^{2} + p^{4} T^{4} \) |
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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
−10.56517336313307916172824570122, −10.02441655292892914162102239351, −9.637227158449665309155269478936, −9.241973325902318881981352711326, −9.030132540461166159497014370537, −8.220171770233155836725799912934, −7.964982167476958877023121179342, −7.42468238953349660119006403105, −6.98600445400844934925746173742, −6.28150930185917610736618840046, −6.09198446715626975543853643375, −5.49256970275830292974944989177, −5.20071365269357860493764637821, −4.61165829364877076659823215537, −4.15191890327175795303595920573, −3.04486548414462012430477027238, −2.70923038642650021044171177521, −2.18193441058448371973173136984, −1.70993225594678463379501864936, −0.34676736843814697764783582293,
0.34676736843814697764783582293, 1.70993225594678463379501864936, 2.18193441058448371973173136984, 2.70923038642650021044171177521, 3.04486548414462012430477027238, 4.15191890327175795303595920573, 4.61165829364877076659823215537, 5.20071365269357860493764637821, 5.49256970275830292974944989177, 6.09198446715626975543853643375, 6.28150930185917610736618840046, 6.98600445400844934925746173742, 7.42468238953349660119006403105, 7.964982167476958877023121179342, 8.220171770233155836725799912934, 9.030132540461166159497014370537, 9.241973325902318881981352711326, 9.637227158449665309155269478936, 10.02441655292892914162102239351, 10.56517336313307916172824570122
