L(s) = 1 | − 0.526·5-s + (2.43 − 1.02i)7-s − 4.61·11-s + (0.244 + 0.423i)13-s + (−2.75 − 4.77i)17-s + (1.83 − 3.18i)19-s + 0.0539·23-s − 4.72·25-s + (3.28 − 5.68i)29-s + (−3.03 + 5.26i)31-s + (−1.28 + 0.538i)35-s + (0.223 − 0.387i)37-s + (−2.52 − 4.36i)41-s + (2.84 − 4.93i)43-s + (−4.59 − 7.96i)47-s + ⋯ |
L(s) = 1 | − 0.235·5-s + (0.922 − 0.386i)7-s − 1.39·11-s + (0.0678 + 0.117i)13-s + (−0.668 − 1.15i)17-s + (0.421 − 0.730i)19-s + 0.0112·23-s − 0.944·25-s + (0.609 − 1.05i)29-s + (−0.545 + 0.945i)31-s + (−0.216 + 0.0910i)35-s + (0.0367 − 0.0637i)37-s + (−0.394 − 0.682i)41-s + (0.434 − 0.752i)43-s + (−0.670 − 1.16i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.019713014\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.019713014\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.43 + 1.02i)T \) |
good | 5 | \( 1 + 0.526T + 5T^{2} \) |
| 11 | \( 1 + 4.61T + 11T^{2} \) |
| 13 | \( 1 + (-0.244 - 0.423i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.75 + 4.77i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.83 + 3.18i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 0.0539T + 23T^{2} \) |
| 29 | \( 1 + (-3.28 + 5.68i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.03 - 5.26i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.223 + 0.387i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.52 + 4.36i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.84 + 4.93i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.59 + 7.96i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.37 - 7.57i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.31 - 5.74i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.232 - 0.403i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.59 - 4.49i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 1.76T + 71T^{2} \) |
| 73 | \( 1 + (5.23 + 9.07i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.18 + 14.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.49 - 7.78i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.05 + 12.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.22 + 9.04i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.091613776878333122885770966124, −8.403115654161467852715156002447, −7.47776369033302289566806297972, −7.14745455349020858778344681157, −5.76380023263860946407994994371, −4.97424134515801866189811204009, −4.31238637165609727551919784264, −3.00885606856853298113080472154, −2.02179872524003875820604720440, −0.38797655614470659273626257688,
1.57937850001283814262744614642, 2.61626757214501390306656699688, 3.81397201708231339574321862051, 4.80861162711176684656147092486, 5.54550150841364269517960482202, 6.37579982469976482496570015333, 7.69878127965195019614430168957, 7.985863092362498362920572495893, 8.750977819975905262742271098872, 9.800719619716761344908587958439