| L(s) = 1 | + 3·2-s − 2·3-s + 4·4-s + 5-s − 6·6-s + 2·7-s + 3·8-s + 3·9-s + 3·10-s − 8·12-s + 4·13-s + 6·14-s − 2·15-s + 3·16-s + 9·17-s + 9·18-s − 5·19-s + 4·20-s − 4·21-s − 4·23-s − 6·24-s − 8·25-s + 12·26-s − 4·27-s + 8·28-s + 12·29-s − 6·30-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.15·3-s + 2·4-s + 0.447·5-s − 2.44·6-s + 0.755·7-s + 1.06·8-s + 9-s + 0.948·10-s − 2.30·12-s + 1.10·13-s + 1.60·14-s − 0.516·15-s + 3/4·16-s + 2.18·17-s + 2.12·18-s − 1.14·19-s + 0.894·20-s − 0.872·21-s − 0.834·23-s − 1.22·24-s − 8/5·25-s + 2.35·26-s − 0.769·27-s + 1.51·28-s + 2.22·29-s − 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.082440162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.082440162\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 43 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 85 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 83 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 107 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 17 T + 189 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 23 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 125 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 87 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 155 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 209 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 203 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−11.61215141032998682664216013847, −11.58919886972212187961866128887, −11.05033757182290375695842507180, −10.25337631368427615182969572735, −10.07821864330822044545877542179, −9.861919358650799954355187995432, −8.641764915832080958808547938776, −8.315313212703412726921943271422, −7.80965097349434811601625074813, −7.16132082040376887648743902098, −6.32134835294888550388313327552, −6.18219234634381625701084691188, −5.63369731786823936784536233854, −5.38132149526540359444223064265, −4.59053258519555098502675020313, −4.51559678261015459462304969752, −3.58051138352353501631455446828, −3.37718679860900972842092947176, −2.04984432008032180258301225430, −1.26179673917819932488302670889,
1.26179673917819932488302670889, 2.04984432008032180258301225430, 3.37718679860900972842092947176, 3.58051138352353501631455446828, 4.51559678261015459462304969752, 4.59053258519555098502675020313, 5.38132149526540359444223064265, 5.63369731786823936784536233854, 6.18219234634381625701084691188, 6.32134835294888550388313327552, 7.16132082040376887648743902098, 7.80965097349434811601625074813, 8.315313212703412726921943271422, 8.641764915832080958808547938776, 9.861919358650799954355187995432, 10.07821864330822044545877542179, 10.25337631368427615182969572735, 11.05033757182290375695842507180, 11.58919886972212187961866128887, 11.61215141032998682664216013847
