L(s) = 1 | − 2-s − 4-s + 3·5-s + 7-s + 3·8-s − 3·10-s + 6·11-s − 6·13-s − 14-s − 16-s − 17-s − 6·19-s − 3·20-s − 6·22-s − 6·23-s + 4·25-s + 6·26-s − 28-s − 4·31-s − 5·32-s + 34-s + 3·35-s − 3·37-s + 6·38-s + 9·40-s − 11·41-s + 11·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.34·5-s + 0.377·7-s + 1.06·8-s − 0.948·10-s + 1.80·11-s − 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.242·17-s − 1.37·19-s − 0.670·20-s − 1.27·22-s − 1.25·23-s + 4/5·25-s + 1.17·26-s − 0.188·28-s − 0.718·31-s − 0.883·32-s + 0.171·34-s + 0.507·35-s − 0.493·37-s + 0.973·38-s + 1.42·40-s − 1.71·41-s + 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.243627138\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.243627138\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.48147799198481, −13.97495419563427, −13.65141501028566, −12.83463579324565, −12.50547019329181, −11.93730403541197, −11.27552188441864, −10.66612958729592, −10.06806376773982, −9.768724887407919, −9.302810186986140, −8.914145440596168, −8.322737388101359, −7.757917747743122, −7.006896363429795, −6.548246822628655, −6.042203498517136, −5.225514594029213, −4.816888063153660, −4.143356604280502, −3.635648703460870, −2.407588447583559, −1.873829190456968, −1.545051627450863, −0.4295774654835194,
0.4295774654835194, 1.545051627450863, 1.873829190456968, 2.407588447583559, 3.635648703460870, 4.143356604280502, 4.816888063153660, 5.225514594029213, 6.042203498517136, 6.548246822628655, 7.006896363429795, 7.757917747743122, 8.322737388101359, 8.914145440596168, 9.302810186986140, 9.768724887407919, 10.06806376773982, 10.66612958729592, 11.27552188441864, 11.93730403541197, 12.50547019329181, 12.83463579324565, 13.65141501028566, 13.97495419563427, 14.48147799198481