L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.5 + 2.59i)5-s + (−0.5 − 2.59i)7-s − 0.999·8-s + (−1.5 + 2.59i)10-s + (1.5 − 2.59i)11-s + 13-s + (2 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (2 + 3.46i)19-s − 3·20-s + 3·22-s + (3 + 5.19i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.670 + 1.16i)5-s + (−0.188 − 0.981i)7-s − 0.353·8-s + (−0.474 + 0.821i)10-s + (0.452 − 0.783i)11-s + 0.277·13-s + (0.534 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (0.458 + 0.794i)19-s − 0.670·20-s + 0.639·22-s + (0.625 + 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.415036546\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.415036546\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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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
−9.677286499342414638979079150218, −8.657381878749197627320158681074, −7.59049832618749718059399906596, −7.09623980941927222602735005639, −6.33032608211884014369495241872, −5.68381845720599514463471482957, −4.62321460050586810059880066670, −3.36747468089217585794957718983, −3.02667421671310850877187875558, −1.17240791147503843095174597719,
1.06755725453847286167940506226, 2.04132312773080753569056477495, 3.06127535050666841314399495950, 4.40262111055043961759429878924, 4.94117735804494608267835584105, 5.89779746362935082912521645757, 6.43074036615804138800314602138, 7.87264543978164493121379191625, 8.774867611674503932879902599317, 9.256641012689256851608678656865