L(s) = 1 | + (−1.92 − 2.07i)2-s − 3i·3-s + (−0.577 + 7.97i)4-s − 14.7i·5-s + (−6.21 + 5.77i)6-s − 7·7-s + (17.6 − 14.1i)8-s − 9·9-s + (−30.5 + 28.3i)10-s − 38.2i·11-s + (23.9 + 1.73i)12-s + 10.4i·13-s + (13.4 + 14.4i)14-s − 44.2·15-s + (−63.3 − 9.21i)16-s − 51.5·17-s + ⋯ |
L(s) = 1 | + (−0.681 − 0.732i)2-s − 0.577i·3-s + (−0.0721 + 0.997i)4-s − 1.31i·5-s + (−0.422 + 0.393i)6-s − 0.377·7-s + (0.779 − 0.626i)8-s − 0.333·9-s + (−0.965 + 0.897i)10-s − 1.04i·11-s + (0.575 + 0.0416i)12-s + 0.222i·13-s + (0.257 + 0.276i)14-s − 0.761·15-s + (−0.989 − 0.143i)16-s − 0.734·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.229392 + 0.478687i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.229392 + 0.478687i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.92 + 2.07i)T \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 + 7T \) |
good | 5 | \( 1 + 14.7iT - 125T^{2} \) |
| 11 | \( 1 + 38.2iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 10.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 51.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4.22iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 8.91T + 1.21e4T^{2} \) |
| 29 | \( 1 - 248. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 197.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 84.7iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 272.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 269. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 547.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 203. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 584. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 840. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 611. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 265.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 158.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 523.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.29e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 157.T + 9.12e5T^{2} \) |
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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
−11.66832995195467373825885793441, −10.85521271616265195884525391585, −9.374673493018022340576387223119, −8.777710434483038312516353974909, −7.890422490843195262649838539996, −6.50442370717775166808088002979, −4.91246325852321983555268882955, −3.35285118337076047888525906092, −1.61676520525144547269637455877, −0.30383524737494982420566318249,
2.41941909671912799677555254225, 4.21181005816229782785510852629, 5.76387714681660250626624881459, 6.81327754793074122852028361918, 7.62218455121241270244373060855, 9.042257592987935312096593373029, 9.978060711747664724014591045264, 10.59590464106126411882504568051, 11.57758777588450759961877437131, 13.21386870398376252812379767339