| L(s) = 1 | + (0.854 − 1.12i)2-s + (−1.36 + 1.06i)3-s + (−0.539 − 1.92i)4-s + (−0.639 − 0.553i)5-s + (0.0414 + 2.44i)6-s + (−1.43 − 2.22i)7-s + (−2.63 − 1.03i)8-s + (0.710 − 2.91i)9-s + (−1.17 + 0.246i)10-s + (−0.134 + 0.934i)11-s + (2.79 + 2.04i)12-s + (−2.85 − 1.83i)13-s + (−3.73 − 0.290i)14-s + (1.46 + 0.0705i)15-s + (−3.41 + 2.07i)16-s + (0.501 − 1.70i)17-s + ⋯ |
| L(s) = 1 | + (0.604 − 0.796i)2-s + (−0.786 + 0.617i)3-s + (−0.269 − 0.962i)4-s + (−0.285 − 0.247i)5-s + (0.0169 + 0.999i)6-s + (−0.540 − 0.841i)7-s + (−0.930 − 0.367i)8-s + (0.236 − 0.971i)9-s + (−0.370 + 0.0780i)10-s + (−0.0405 + 0.281i)11-s + (0.806 + 0.590i)12-s + (−0.792 − 0.509i)13-s + (−0.997 − 0.0776i)14-s + (0.377 + 0.0182i)15-s + (−0.854 + 0.519i)16-s + (0.121 − 0.414i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.216451 - 0.824653i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.216451 - 0.824653i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.854 + 1.12i)T \) |
| 3 | \( 1 + (1.36 - 1.06i)T \) |
| 23 | \( 1 + (-2.16 - 4.27i)T \) |
| good | 5 | \( 1 + (0.639 + 0.553i)T + (0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (1.43 + 2.22i)T + (-2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.134 - 0.934i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (2.85 + 1.83i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.501 + 1.70i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (1.49 + 5.10i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (0.474 - 1.61i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-4.47 + 2.04i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-6.10 - 7.04i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (4.13 + 3.57i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.32 - 1.97i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + (-2.34 - 3.64i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (8.58 + 5.51i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (2.08 + 4.56i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (9.29 - 1.33i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.829 + 5.76i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-5.76 + 1.69i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-2.39 + 3.72i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (3.69 + 4.26i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (8.79 + 4.01i)T + (58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-7.68 + 8.87i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.53254976957644352955567613026, −10.60108674844289194358629796859, −9.947708210522799016224528071586, −9.128395736371169921941152891524, −7.31093686715686904259307368572, −6.22011775582843479087068564912, −4.96680588660275753455198878941, −4.28688996721029623104590037228, −3.00287951811630316238133855000, −0.58007709371819631553269112432,
2.59775546627187867654463472401, 4.22919732434381444374622957957, 5.53657855320497919452178857583, 6.22067185623452651399939666056, 7.16928993078783982036201498995, 8.080852501945638468789960585293, 9.207127926936445918421433336337, 10.60531770453920458267388183994, 11.79934915789136374558741799436, 12.33248496454494588601683263039
