L(s) = 1 | + (−0.673 + 1.16i)2-s + (1.70 + 0.300i)3-s + (0.0923 + 0.160i)4-s + (−1.26 − 2.19i)5-s + (−1.49 + 1.78i)6-s + (−0.5 + 0.866i)7-s − 2.94·8-s + (2.81 + 1.02i)9-s + 3.41·10-s + (−0.233 + 0.405i)11-s + (0.109 + 0.300i)12-s + (−2.91 − 5.04i)13-s + (−0.673 − 1.16i)14-s + (−1.5 − 4.12i)15-s + (1.79 − 3.11i)16-s + 3.87·17-s + ⋯ |
L(s) = 1 | + (−0.476 + 0.825i)2-s + (0.984 + 0.173i)3-s + (0.0461 + 0.0800i)4-s + (−0.566 − 0.980i)5-s + (−0.612 + 0.729i)6-s + (−0.188 + 0.327i)7-s − 1.04·8-s + (0.939 + 0.342i)9-s + 1.07·10-s + (−0.0705 + 0.122i)11-s + (0.0316 + 0.0868i)12-s + (−0.807 − 1.39i)13-s + (−0.180 − 0.311i)14-s + (−0.387 − 1.06i)15-s + (0.449 − 0.778i)16-s + 0.940·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.754133 + 0.435399i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.754133 + 0.435399i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.70 - 0.300i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.673 - 1.16i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.26 + 2.19i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.233 - 0.405i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.91 + 5.04i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 19 | \( 1 + 2.18T + 19T^{2} \) |
| 23 | \( 1 + (-0.0530 - 0.0918i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.39 - 7.60i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.84 - 6.65i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.68T + 37T^{2} \) |
| 41 | \( 1 + (-1.11 - 1.92i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.613 - 1.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.66 + 4.61i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.716T + 53T^{2} \) |
| 59 | \( 1 + (0.368 + 0.637i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.479 - 0.829i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.81 - 8.34i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + (-6.31 + 10.9i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.36 + 2.36i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 8.11T + 89T^{2} \) |
| 97 | \( 1 + (-6.80 + 11.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−15.37584204586849953575466606069, −14.51049208976836894755538779096, −12.76706954420576090456725363213, −12.28916140940002607108971259968, −10.16131892573071792915809126011, −8.876095289468611857697051455483, −8.188695489686515485120025030087, −7.22735312914029273535233301002, −5.19788020819904888395008645238, −3.22511432294268499379528251578,
2.34767761225658643415147265346, 3.77827692863151457379500076016, 6.60933675217119692794223740061, 7.75112478353975874046891473992, 9.300025567364884638193498655487, 10.13179156561009136394893239268, 11.31863124537251123039268213012, 12.31010736905681270687955969454, 13.88027788009327946045370078592, 14.74136465303592868309824437185