L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.923 − 1.60i)5-s − 0.999i·8-s + (−1.60 − 0.923i)10-s + 0.765i·13-s + (−0.5 − 0.866i)16-s + (0.382 − 0.662i)17-s − 1.84·20-s + (−1.20 + 2.09i)25-s + (0.382 + 0.662i)26-s + (−0.866 − 0.499i)32-s − 0.765i·34-s + (−0.707 − 1.22i)37-s + (−1.60 + 0.923i)40-s − 0.765·41-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.923 − 1.60i)5-s − 0.999i·8-s + (−1.60 − 0.923i)10-s + 0.765i·13-s + (−0.5 − 0.866i)16-s + (0.382 − 0.662i)17-s − 1.84·20-s + (−1.20 + 2.09i)25-s + (0.382 + 0.662i)26-s + (−0.866 − 0.499i)32-s − 0.765i·34-s + (−0.707 − 1.22i)37-s + (−1.60 + 0.923i)40-s − 0.765·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.503480020\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.503480020\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 0.765iT - T^{2} \) |
| 17 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + 0.765T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.60 + 0.923i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - 1.84iT - T^{2} \) |
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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
−9.220420642807406227430504707823, −8.529933684671257888618101916460, −7.55706841544811138218050440988, −6.76865929761692387931830169831, −5.51522205598165557853560080906, −5.01440499330717945606602644489, −4.17280273572742934561475699948, −3.55781220729518562522688689317, −2.09158838860886240809228948874, −0.877955672697376951520529859658,
2.32722492502241117114289205460, 3.33287072407003277890059011957, 3.71294901667583225909262881383, 4.87772856695002523657965067371, 5.90063704736479115758730900900, 6.64167428006915078841579938634, 7.26307299380992305988095806196, 7.963337988475289814993052690168, 8.568887003604623103621677740967, 10.17610673070879892125960533014