L(s) = 1 | − 0.879·2-s − 1.22·4-s + 1.34·5-s + 7-s + 2.83·8-s − 1.18·10-s + 1.65·11-s − 3.36·13-s − 0.879·14-s − 0.0418·16-s + 0.467·17-s − 3.22·19-s − 1.65·20-s − 1.45·22-s + 8.94·23-s − 3.18·25-s + 2.96·26-s − 1.22·28-s + 6.26·29-s + 9.23·31-s − 5.63·32-s − 0.411·34-s + 1.34·35-s + 9.23·37-s + 2.83·38-s + 3.82·40-s + 3.41·41-s + ⋯ |
L(s) = 1 | − 0.621·2-s − 0.613·4-s + 0.602·5-s + 0.377·7-s + 1.00·8-s − 0.374·10-s + 0.498·11-s − 0.934·13-s − 0.235·14-s − 0.0104·16-s + 0.113·17-s − 0.740·19-s − 0.369·20-s − 0.309·22-s + 1.86·23-s − 0.636·25-s + 0.581·26-s − 0.231·28-s + 1.16·29-s + 1.65·31-s − 0.996·32-s − 0.0705·34-s + 0.227·35-s + 1.51·37-s + 0.460·38-s + 0.604·40-s + 0.532·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.057615605\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.057615605\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 + 0.879T + 2T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 - 0.467T + 17T^{2} \) |
| 19 | \( 1 + 3.22T + 19T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 - 6.26T + 29T^{2} \) |
| 31 | \( 1 - 9.23T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 - 3.41T + 41T^{2} \) |
| 43 | \( 1 + 4.41T + 43T^{2} \) |
| 47 | \( 1 - 9.35T + 47T^{2} \) |
| 53 | \( 1 + 0.573T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 7.63T + 61T^{2} \) |
| 67 | \( 1 - 0.596T + 67T^{2} \) |
| 71 | \( 1 + 0.554T + 71T^{2} \) |
| 73 | \( 1 - 2.04T + 73T^{2} \) |
| 79 | \( 1 + 2.40T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 9.08T + 89T^{2} \) |
| 97 | \( 1 + 1.89T + 97T^{2} \) |
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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
−10.46219581327081445768060215777, −9.787775767455508384753753324705, −9.065158079132529498758868610467, −8.270944694888021880510815649831, −7.33668117130090367415398609845, −6.26475028854853570254533829376, −5.02363653939044325409532213645, −4.31305047020691602191174489411, −2.59986496468464481873497793337, −1.07966512384365140624752892278,
1.07966512384365140624752892278, 2.59986496468464481873497793337, 4.31305047020691602191174489411, 5.02363653939044325409532213645, 6.26475028854853570254533829376, 7.33668117130090367415398609845, 8.270944694888021880510815649831, 9.065158079132529498758868610467, 9.787775767455508384753753324705, 10.46219581327081445768060215777