| L(s) = 1 | − 4·5-s + 9-s + 4·13-s − 12·17-s + 2·25-s − 4·29-s + 4·37-s + 4·41-s − 4·45-s + 2·49-s − 20·53-s − 12·61-s − 16·65-s − 12·73-s + 81-s + 48·85-s + 20·89-s − 28·97-s + 12·101-s − 28·109-s + 4·113-s + 4·117-s − 6·121-s + 28·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1/3·9-s + 1.10·13-s − 2.91·17-s + 2/5·25-s − 0.742·29-s + 0.657·37-s + 0.624·41-s − 0.596·45-s + 2/7·49-s − 2.74·53-s − 1.53·61-s − 1.98·65-s − 1.40·73-s + 1/9·81-s + 5.20·85-s + 2.11·89-s − 2.84·97-s + 1.19·101-s − 2.68·109-s + 0.376·113-s + 0.369·117-s − 0.545·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−10.20586918892848667331432436971, −9.240123327709471504547415939693, −9.133901360741508328144538474134, −8.460175078145674768211752918036, −7.83182394757322498939219938639, −7.67780945046588747720337476950, −6.80946897903754876349072202165, −6.44896944953500655110243120870, −5.77092473843782707280190480124, −4.62570703678849832502580328887, −4.31158789218116792910899233548, −3.85566973953101374011358603281, −3.00571163091752467289605693825, −1.83689405023894510739841616557, 0,
1.83689405023894510739841616557, 3.00571163091752467289605693825, 3.85566973953101374011358603281, 4.31158789218116792910899233548, 4.62570703678849832502580328887, 5.77092473843782707280190480124, 6.44896944953500655110243120870, 6.80946897903754876349072202165, 7.67780945046588747720337476950, 7.83182394757322498939219938639, 8.460175078145674768211752918036, 9.133901360741508328144538474134, 9.240123327709471504547415939693, 10.20586918892848667331432436971
