L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−2.14 + 0.618i)5-s + 3.53i·7-s − 0.999i·8-s + (−1.55 + 1.60i)10-s − 3.34·11-s + (−1.48 − 0.854i)13-s + (1.76 + 3.06i)14-s + (−0.5 − 0.866i)16-s + (3.29 − 1.90i)17-s + (1.30 − 4.15i)19-s + (−0.539 + 2.17i)20-s + (−2.89 + 1.67i)22-s + (−7.33 − 4.23i)23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.961 + 0.276i)5-s + 1.33i·7-s − 0.353i·8-s + (−0.490 + 0.509i)10-s − 1.00·11-s + (−0.410 − 0.237i)13-s + (0.472 + 0.819i)14-s + (−0.125 − 0.216i)16-s + (0.798 − 0.460i)17-s + (0.300 − 0.953i)19-s + (−0.120 + 0.485i)20-s + (−0.616 + 0.356i)22-s + (−1.52 − 0.882i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.689 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9119516326\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9119516326\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (2.14 - 0.618i)T \) |
| 19 | \( 1 + (-1.30 + 4.15i)T \) |
good | 7 | \( 1 - 3.53iT - 7T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 13 | \( 1 + (1.48 + 0.854i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.29 + 1.90i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (7.33 + 4.23i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.97 + 6.89i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.07T + 31T^{2} \) |
| 37 | \( 1 + 4.70iT - 37T^{2} \) |
| 41 | \( 1 + (5.96 + 10.3i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.39 - 2.53i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.73 + 2.73i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.8 - 6.25i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.93 + 5.08i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.92 - 11.9i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.82 - 4.51i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.04 + 7.00i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.11 + 2.95i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.53 + 6.13i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.68iT - 83T^{2} \) |
| 89 | \( 1 + (4.66 - 8.07i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.06 + 2.92i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.994268055107030643252390756381, −8.171356407382442547110655764328, −7.53051995728556979284754439542, −6.51255876963795782601891856424, −5.56871332401840335695054925347, −4.95725945853935891688717162705, −3.96355169879947341269309425946, −2.81324716462591959709735815705, −2.38168008145033846846589402729, −0.27726018949722587497781034509,
1.43553961211270615507582537138, 3.19775204340177783137700312297, 3.77929664070759190698427586844, 4.66794558396593078021825066719, 5.37389866876420070657447546186, 6.52285642665319347058014283530, 7.33833900280072870657489676118, 7.957319766483979444129843549918, 8.330001872366016447451781376333, 9.935638334451453039768704490638