L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (0.499 + 0.866i)6-s + (−2.59 + 1.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s − 0.999i·12-s + (−2.59 − 2.5i)13-s + 3·14-s + (−0.5 + 0.866i)16-s − 0.999i·18-s + (−2.5 − 4.33i)19-s + 3·21-s + (0.866 − 0.499i)22-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.204 + 0.353i)6-s + (−0.981 + 0.566i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.150 + 0.261i)11-s − 0.288i·12-s + (−0.720 − 0.693i)13-s + 0.801·14-s + (−0.125 + 0.216i)16-s − 0.235i·18-s + (−0.573 − 0.993i)19-s + 0.654·21-s + (0.184 − 0.106i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6334139420\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6334139420\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (2.59 + 2.5i)T \) |
good | 7 | \( 1 + (2.59 - 1.5i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.46 + 2i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.73 + i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 - 13iT - 53T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.3 - 6i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1 - 1.73i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.3 + 6i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.267738277901221500456561268117, −8.476315191616653862554376265165, −7.69643307866001253905120533640, −6.78399374201839204982682837579, −6.24281797698780537196447559289, −5.24288394262853844080481793506, −4.25954523239048061148651515877, −2.90671106654086169811173202293, −2.32851210680820132126305005842, −0.69313364994574397675756963150,
0.47984753263988663761659026834, 2.02116409797950001818676701295, 3.38218667500963820800756458191, 4.30005960285125334388322367736, 5.27757529334371296657821366620, 6.34722435401084913572718038489, 6.59436001982230552700551867810, 7.64779184478706476186580784854, 8.333556504078839047154893690067, 9.421562443698011535640753348800