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 + (−1.73 − i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (2.5 + 4.33i)11-s + 0.999i·12-s + (−3.46 − i)13-s − 1.99·14-s + (−0.5 − 0.866i)16-s + (−1.73 − i)17-s − 0.999i·18-s + (−1 + 1.73i)19-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 + (−0.654 − 0.377i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.753 + 1.30i)11-s + 0.288i·12-s + (−0.960 − 0.277i)13-s − 0.534·14-s + (−0.125 − 0.216i)16-s + (−0.420 − 0.242i)17-s − 0.235i·18-s + (−0.229 + 0.397i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7049284391\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7049284391\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (3.46 + i)T \) |
good | 7 | \( 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 11T + 31T^{2} \) |
| 37 | \( 1 + (2.59 - 1.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.52 - 5.5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + (7.5 - 12.9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (13.8 - 8i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.73 + i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.525064258538017061396171610931, −9.061417411483040065867642892961, −7.46061015395356919352567465008, −7.04597521530385100724488584323, −6.18384023069064505850465277195, −5.28457297368841068058511503464, −4.46293730273541636729741416599, −3.83444926623896607947559928947, −2.70326902081056409607779985780, −1.51260533180443386977805239735,
0.20962138138586612378228140035, 2.00283536386392535551068372740, 3.11107804089958557859208640472, 4.03454564218917444515837502544, 4.99424847125880081981777695669, 5.94449468473898000115201957124, 6.32800109421215922739135478849, 7.15902617652239968563065554869, 7.971477712833679421797284325377, 9.018377814117975807433746457580