L(s) = 1 | + (−0.861 + 0.129i)2-s + (−1.18 + 0.365i)4-s + (−0.317 + 4.23i)5-s + (2.28 − 1.33i)7-s + (2.54 − 1.22i)8-s + (−0.276 − 3.68i)10-s + (−0.0107 + 0.0275i)11-s + (1.53 + 1.92i)13-s + (−1.79 + 1.44i)14-s + (0.0179 − 0.0122i)16-s + (−3.29 + 3.05i)17-s + (−2.70 + 4.68i)19-s + (−1.17 − 5.13i)20-s + (0.00573 − 0.0251i)22-s + (−2.28 − 2.11i)23-s + ⋯ |
L(s) = 1 | + (−0.609 + 0.0918i)2-s + (−0.592 + 0.182i)4-s + (−0.141 + 1.89i)5-s + (0.863 − 0.504i)7-s + (0.899 − 0.433i)8-s + (−0.0874 − 1.16i)10-s + (−0.00325 + 0.00829i)11-s + (0.425 + 0.533i)13-s + (−0.479 + 0.386i)14-s + (0.00448 − 0.00305i)16-s + (−0.799 + 0.741i)17-s + (−0.620 + 1.07i)19-s + (−0.262 − 1.14i)20-s + (0.00122 − 0.00535i)22-s + (−0.475 − 0.441i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.267306 + 0.671190i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.267306 + 0.671190i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.28 + 1.33i)T \) |
good | 2 | \( 1 + (0.861 - 0.129i)T + (1.91 - 0.589i)T^{2} \) |
| 5 | \( 1 + (0.317 - 4.23i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (0.0107 - 0.0275i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-1.53 - 1.92i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (3.29 - 3.05i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.70 - 4.68i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.28 + 2.11i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.662 + 2.90i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-1.83 - 3.17i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.34 + 0.414i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (9.33 - 4.49i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-7.47 - 3.60i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-0.583 + 0.0879i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-8.64 + 2.66i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.267 - 3.57i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (12.0 + 3.73i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-1.28 - 2.23i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.49 + 6.55i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-8.56 - 1.29i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (4.05 - 7.02i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.95 + 2.45i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-1.65 - 4.20i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + 9.08T + 97T^{2} \) |
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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
−11.05042373799075265106750858623, −10.56380107167713904810947033487, −9.886946754518182014702675699439, −8.535290102448452351556100577805, −7.896209839084173483696695693613, −6.98947034268421268177423367868, −6.15338127303557338488506221448, −4.38860531229714677456254017974, −3.63424485172204670425416650455, −1.95067918762920997522566952737,
0.58425659822016528638594571913, 1.88739729025936126834464466722, 4.24556933579039665728955105367, 4.93706695939416142092142051702, 5.64890304169116131369161448947, 7.52415476164971056613508337365, 8.528886887203552761549668798125, 8.775439290744103767031154594307, 9.525743551486853601955174159091, 10.74219942663387725690657447686