L(s) = 1 | + (1.29 − 1.19i)2-s + (0.0828 − 1.10i)4-s + (−1.35 + 3.45i)5-s + (1.47 − 2.19i)7-s + (0.979 + 1.22i)8-s + (2.38 + 6.08i)10-s + (1.46 + 0.451i)11-s + (0.545 + 2.39i)13-s + (−0.727 − 4.60i)14-s + (4.93 + 0.743i)16-s + (3.98 − 2.71i)17-s + (1.90 + 3.30i)19-s + (3.70 + 1.78i)20-s + (2.43 − 1.17i)22-s + (−3.98 − 2.71i)23-s + ⋯ |
L(s) = 1 | + (0.913 − 0.848i)2-s + (0.0414 − 0.552i)4-s + (−0.605 + 1.54i)5-s + (0.557 − 0.830i)7-s + (0.346 + 0.434i)8-s + (0.755 + 1.92i)10-s + (0.441 + 0.136i)11-s + (0.151 + 0.662i)13-s + (−0.194 − 1.23i)14-s + (1.23 + 0.185i)16-s + (0.967 − 0.659i)17-s + (0.437 + 0.757i)19-s + (0.828 + 0.398i)20-s + (0.518 − 0.249i)22-s + (−0.830 − 0.566i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.22075 - 0.103205i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.22075 - 0.103205i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.47 + 2.19i)T \) |
good | 2 | \( 1 + (-1.29 + 1.19i)T + (0.149 - 1.99i)T^{2} \) |
| 5 | \( 1 + (1.35 - 3.45i)T + (-3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (-1.46 - 0.451i)T + (9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (-0.545 - 2.39i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-3.98 + 2.71i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-1.90 - 3.30i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.98 + 2.71i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (0.292 + 0.140i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (3.32 - 5.75i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.669 + 8.92i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (1.52 + 1.90i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (3.39 - 4.25i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (6.44 - 5.98i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.754 + 10.0i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (4.97 + 12.6i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-0.0620 - 0.827i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-6.41 + 11.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.993 - 0.478i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (4.14 + 3.84i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-5.85 - 10.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.63 - 7.15i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-15.7 + 4.86i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−11.24805242069727474480021279808, −10.60959244047784975421762923768, −9.791500003595714227365353108155, −8.060382699765374613866891093950, −7.40703220431842486555900746150, −6.44238840807336843307391286382, −4.98854247115031328661482644156, −3.82378632560176619263782056977, −3.34908810952215361833649405175, −1.89304839272661931121185752582,
1.29560726730652720735994798453, 3.58896720534088614777732649863, 4.62701885476791253788298035700, 5.37653095321227707476968576592, 6.02399363204439064036465224013, 7.55073445940031935799569974135, 8.197602504166930329924625513192, 9.043625033109089744822863633066, 10.12463123568646318018255833219, 11.66198157522882744268336418931