| L(s) = 1 | + 2·5-s − 6·9-s + 14·17-s − 7·25-s − 12·45-s − 5·49-s − 30·61-s + 22·73-s + 27·81-s + 28·85-s + 20·101-s + 3·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 84·153-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2·9-s + 3.39·17-s − 7/5·25-s − 1.78·45-s − 5/7·49-s − 3.84·61-s + 2.57·73-s + 3·81-s + 3.03·85-s + 1.99·101-s + 3/11·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 6.79·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.014351525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.014351525\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−9.733159759828253484517382507585, −9.598640470651873021927810079484, −9.289689289756409509468176085632, −8.730652112905267376001911698192, −8.182529881711780159313974762446, −7.892461696995965956192970964650, −7.72824453972356422499019613261, −7.14724539298870820808912222030, −6.27600795084745381564055865545, −6.05933519784711203569568899208, −5.84570053584007553397729896467, −5.27621839607647452944859918194, −5.19137696551245104377824290314, −4.36506464712612395828591044825, −3.44894845950579846685311982810, −3.33993944117661109114552992065, −2.88082250231297329570457088281, −2.10492350651637610444647579425, −1.54550165955808104194140134970, −0.61090724662325347051808232074,
0.61090724662325347051808232074, 1.54550165955808104194140134970, 2.10492350651637610444647579425, 2.88082250231297329570457088281, 3.33993944117661109114552992065, 3.44894845950579846685311982810, 4.36506464712612395828591044825, 5.19137696551245104377824290314, 5.27621839607647452944859918194, 5.84570053584007553397729896467, 6.05933519784711203569568899208, 6.27600795084745381564055865545, 7.14724539298870820808912222030, 7.72824453972356422499019613261, 7.892461696995965956192970964650, 8.182529881711780159313974762446, 8.730652112905267376001911698192, 9.289689289756409509468176085632, 9.598640470651873021927810079484, 9.733159759828253484517382507585
