L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 − 2.59i)7-s − 0.999·8-s + (−0.499 − 0.866i)10-s + (2.5 + 4.33i)11-s + (−2 − 1.73i)14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + (−4 + 6.92i)19-s − 0.999·20-s + 5·22-s + (−2 + 3.46i)23-s + (2 + 3.46i)25-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + (0.188 − 0.981i)7-s − 0.353·8-s + (−0.158 − 0.273i)10-s + (0.753 + 1.30i)11-s + (−0.534 − 0.462i)14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + (−0.917 + 1.58i)19-s − 0.223·20-s + 1.06·22-s + (−0.417 + 0.722i)23-s + (0.400 + 0.692i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06336 - 0.707334i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06336 - 0.707334i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 + 2.59i)T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.5 + 9.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.5 - 2.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−13.05919227638037664297531762405, −12.26729944209883985860924735961, −11.19804387518228870911849833650, −10.08867544167712863555511615087, −9.319214638126003556289231861894, −7.76856751447416120519614537958, −6.49441517147219400920310142279, −4.86508935479929062219202311669, −3.87326311241897651525611803845, −1.71549908564854784200259182748,
2.77065576506767648139660305239, 4.49733698997504881248109383069, 6.00841702721639390759489551450, 6.64320906473599423235485103128, 8.446824154179886046524648322759, 8.906846937294824507827450763507, 10.60673168782518327935907196944, 11.61305854131731947052697741727, 12.68554546318390037444242149728, 13.73666016082185373524680323287