| L(s) = 1 | − 2·2-s − 4-s + 4·5-s + 8·8-s − 2·9-s − 8·10-s − 7·16-s + 4·18-s − 4·20-s + 2·25-s − 16·31-s − 14·32-s + 2·36-s + 4·37-s + 32·40-s − 6·41-s + 8·43-s − 8·45-s + 6·49-s − 4·50-s + 8·59-s − 4·61-s + 32·62-s + 35·64-s − 16·72-s + 28·73-s − 8·74-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 1.78·5-s + 2.82·8-s − 2/3·9-s − 2.52·10-s − 7/4·16-s + 0.942·18-s − 0.894·20-s + 2/5·25-s − 2.87·31-s − 2.47·32-s + 1/3·36-s + 0.657·37-s + 5.05·40-s − 0.937·41-s + 1.21·43-s − 1.19·45-s + 6/7·49-s − 0.565·50-s + 1.04·59-s − 0.512·61-s + 4.06·62-s + 35/8·64-s − 1.88·72-s + 3.27·73-s − 0.929·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3227510337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3227510337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 41 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.66809253979925475129188192358, −16.45090550021354385961548599680, −15.31327332951823138445492613296, −14.27037920205550420540120227230, −14.21015703519129594402689683024, −13.59086696649971573679600710911, −13.06769798423458691737584804664, −12.57803490260129489889326697106, −11.27127543477128568491703033953, −10.70342936986858428210829384161, −10.08410493619036205752481581678, −9.444111107550427092499659720523, −9.292152171056600663713839544008, −8.580985768337373898093317880719, −7.86311677130305900610425140845, −7.01517752492680841372861233894, −5.66865630614454274013827923632, −5.37581885900989168966656213824, −4.01466413952758625440605420673, −1.91567106149496012342088715563,
1.91567106149496012342088715563, 4.01466413952758625440605420673, 5.37581885900989168966656213824, 5.66865630614454274013827923632, 7.01517752492680841372861233894, 7.86311677130305900610425140845, 8.580985768337373898093317880719, 9.292152171056600663713839544008, 9.444111107550427092499659720523, 10.08410493619036205752481581678, 10.70342936986858428210829384161, 11.27127543477128568491703033953, 12.57803490260129489889326697106, 13.06769798423458691737584804664, 13.59086696649971573679600710911, 14.21015703519129594402689683024, 14.27037920205550420540120227230, 15.31327332951823138445492613296, 16.45090550021354385961548599680, 16.66809253979925475129188192358
