L(s) = 1 | + (1.06 + 0.926i)2-s + (1.90 − 1.10i)3-s + (0.281 + 1.98i)4-s + (0.5 − 0.866i)5-s + (3.05 + 0.591i)6-s + (0.584 + 2.58i)7-s + (−1.53 + 2.37i)8-s + (0.922 − 1.59i)9-s + (1.33 − 0.461i)10-s + (−2.90 − 5.03i)11-s + (2.71 + 3.46i)12-s − 4.83·13-s + (−1.76 + 3.29i)14-s − 2.20i·15-s + (−3.84 + 1.11i)16-s + (3.78 − 2.18i)17-s + ⋯ |
L(s) = 1 | + (0.755 + 0.655i)2-s + (1.10 − 0.635i)3-s + (0.140 + 0.990i)4-s + (0.223 − 0.387i)5-s + (1.24 + 0.241i)6-s + (0.220 + 0.975i)7-s + (−0.542 + 0.840i)8-s + (0.307 − 0.532i)9-s + (0.422 − 0.145i)10-s + (−0.875 − 1.51i)11-s + (0.784 + 0.999i)12-s − 1.34·13-s + (−0.472 + 0.881i)14-s − 0.568i·15-s + (−0.960 + 0.279i)16-s + (0.917 − 0.529i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.38221 + 0.697861i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.38221 + 0.697861i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.06 - 0.926i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.584 - 2.58i)T \) |
good | 3 | \( 1 + (-1.90 + 1.10i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (2.90 + 5.03i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.83T + 13T^{2} \) |
| 17 | \( 1 + (-3.78 + 2.18i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.63 - 0.945i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.157 + 0.0911i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.38iT - 29T^{2} \) |
| 31 | \( 1 + (-2.03 - 3.53i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.69 + 2.13i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.44iT - 41T^{2} \) |
| 43 | \( 1 + 2.10T + 43T^{2} \) |
| 47 | \( 1 + (0.946 - 1.63i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.54 + 4.93i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.35 - 3.66i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.89 + 8.47i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.04 - 10.4i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.80iT - 71T^{2} \) |
| 73 | \( 1 + (-7.34 + 4.24i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (10.5 + 6.09i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 16.9iT - 83T^{2} \) |
| 89 | \( 1 + (5.11 + 2.95i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 13.2iT - 97T^{2} \) |
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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
−12.31115218171866277483953697236, −11.43724857424098103533683128087, −9.722882615946267437858658845273, −8.552157048394153547716652737748, −8.141370767905101533845288673105, −7.21184010463788570796754248618, −5.74593730723812677711729608909, −5.08785115953684973434899949826, −3.19201072659586267901510387329, −2.42777943088373480441538426415,
2.12344675773668501207039602254, 3.20903686494542219535484784038, 4.32889459366383985420056812152, 5.20313561279848941475356847784, 6.98198274783155421432613518338, 7.79115980727282689514901257460, 9.396935833327469244764006397054, 10.13479660558828603257620273513, 10.43188897980570000264729325156, 11.92043632591920973582219277036