L(s) = 1 | + (−0.132 − 0.229i)2-s + (1.45 − 2.51i)3-s + (0.964 − 1.67i)4-s + (−0.717 − 1.24i)5-s − 0.769·6-s − 1.03·8-s + (−2.73 − 4.73i)9-s + (−0.189 + 0.328i)10-s + (−2.75 + 4.76i)11-s + (−2.80 − 4.86i)12-s − 13-s − 4.17·15-s + (−1.79 − 3.10i)16-s + (2.41 − 4.18i)17-s + (−0.722 + 1.25i)18-s + (1.41 + 2.44i)19-s + ⋯ |
L(s) = 1 | + (−0.0935 − 0.162i)2-s + (0.839 − 1.45i)3-s + (0.482 − 0.835i)4-s + (−0.320 − 0.555i)5-s − 0.314·6-s − 0.367·8-s + (−0.910 − 1.57i)9-s + (−0.0600 + 0.104i)10-s + (−0.829 + 1.43i)11-s + (−0.810 − 1.40i)12-s − 0.277·13-s − 1.07·15-s + (−0.448 − 0.776i)16-s + (0.585 − 1.01i)17-s + (−0.170 + 0.295i)18-s + (0.323 + 0.560i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.283680 - 1.73456i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.283680 - 1.73456i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + (0.132 + 0.229i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.45 + 2.51i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.717 + 1.24i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.75 - 4.76i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.41 + 4.18i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.99 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.04T + 29T^{2} \) |
| 31 | \( 1 + (-4.60 + 7.97i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.306 + 0.530i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 8.43T + 43T^{2} \) |
| 47 | \( 1 + (-1.20 - 2.08i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.914 + 1.58i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.435 - 0.754i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.66 + 2.88i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.31 + 5.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.85T + 71T^{2} \) |
| 73 | \( 1 + (-1.57 + 2.72i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.78 - 15.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + (0.497 + 0.861i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−9.872479488697604737173516560567, −9.472702728691315638680248582441, −8.191074979375242046349419177395, −7.49703917631162851758779050304, −6.96159158066853666784036955359, −5.76053618575762177922543881669, −4.72665787897811243499847144843, −2.89711899091391201426487148685, −2.05109008566370431909845023807, −0.901102334752005866011637296960,
2.92159823608996315984967300542, 3.07265137464017317970966950054, 4.18568873221230792616114106612, 5.35814761682224137950660173257, 6.63335379523641607901984979996, 7.77220133842127776106111422981, 8.457636547255583003169658358700, 8.983621929688269994492001062833, 10.31698036940793143886941259988, 10.73050500409332091393694577257