L(s) = 1 | + (0.5 − 0.133i)2-s + (0.866 + 0.5i)3-s + (−1.5 + 0.866i)4-s + (−1.36 + 1.36i)5-s + (0.5 + 0.133i)6-s + (−0.5 + 2.59i)7-s + (−1.36 + 1.36i)8-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−1 − 3.73i)11-s − 1.73·12-s + (−3.23 + 1.59i)13-s + (0.0980 + 1.36i)14-s + (−1.86 + 0.499i)15-s + (1.23 − 2.13i)16-s + (2.59 + 4.5i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.0947i)2-s + (0.499 + 0.288i)3-s + (−0.750 + 0.433i)4-s + (−0.610 + 0.610i)5-s + (0.204 + 0.0546i)6-s + (−0.188 + 0.981i)7-s + (−0.482 + 0.482i)8-s + (0.166 + 0.288i)9-s + (−0.158 + 0.273i)10-s + (−0.301 − 1.12i)11-s − 0.499·12-s + (−0.896 + 0.443i)13-s + (0.0262 + 0.365i)14-s + (−0.481 + 0.129i)15-s + (0.308 − 0.533i)16-s + (0.630 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.681775 + 0.919590i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.681775 + 0.919590i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
| 13 | \( 1 + (3.23 - 1.59i)T \) |
good | 2 | \( 1 + (-0.5 + 0.133i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (1.36 - 1.36i)T - 5iT^{2} \) |
| 11 | \( 1 + (1 + 3.73i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.59 - 4.5i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.73 - 0.732i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-5.36 - 3.09i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.26 + 3.26i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.5 - 9.33i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.40 + 8.96i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (6.92 - 4i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.53 - 2.53i)T + 47iT^{2} \) |
| 53 | \( 1 - 7.73T + 53T^{2} \) |
| 59 | \( 1 + (-0.0980 + 0.366i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (5.59 - 3.23i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.5 - 2.83i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.46 + 5.46i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (8.29 + 8.29i)T + 73iT^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + (6.92 - 6.92i)T - 83iT^{2} \) |
| 89 | \( 1 + (-9.83 + 2.63i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-5.09 - 1.36i)T + (84.0 + 48.5i)T^{2} \) |
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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
−12.06378403291760379979699541402, −11.55563201122025112700316137994, −10.19213256173903311939676726505, −9.214778988571515657430801506565, −8.363442226877048499400720525586, −7.59468128346640574450650601103, −5.99343051456740077056657103395, −4.88449209778668660097914292665, −3.53503368267853552423969768511, −2.84618419016498308683060636612,
0.789626125133881865378522956812, 3.13154243827568743145603366680, 4.53983118649840420943923564850, 5.05704137221555774631937719706, 6.92296017385031215098181482953, 7.60274755376738301156577141312, 8.765177413716485255007008230018, 9.734087679330049217925085853345, 10.39615607617046091474996973223, 12.04457784034661395621884385899