L(s) = 1 | + (0.401 − 0.969i)3-s + (3.49 − 1.44i)5-s + (−3.21 + 3.21i)7-s + (1.34 + 1.34i)9-s + (−0.363 − 0.878i)11-s + (2.37 + 0.985i)13-s − 3.97i·15-s − 1.34i·17-s + (3.10 + 1.28i)19-s + (1.82 + 4.41i)21-s + (4.30 + 4.30i)23-s + (6.60 − 6.60i)25-s + (4.74 − 1.96i)27-s + (−0.600 + 1.44i)29-s + 3.69·31-s + ⋯ |
L(s) = 1 | + (0.231 − 0.559i)3-s + (1.56 − 0.648i)5-s + (−1.21 + 1.21i)7-s + (0.447 + 0.447i)9-s + (−0.109 − 0.264i)11-s + (0.659 + 0.273i)13-s − 1.02i·15-s − 0.326i·17-s + (0.712 + 0.295i)19-s + (0.398 + 0.963i)21-s + (0.896 + 0.896i)23-s + (1.32 − 1.32i)25-s + (0.914 − 0.378i)27-s + (−0.111 + 0.269i)29-s + 0.663·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.196719450\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.196719450\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-0.401 + 0.969i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-3.49 + 1.44i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (3.21 - 3.21i)T - 7iT^{2} \) |
| 11 | \( 1 + (0.363 + 0.878i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-2.37 - 0.985i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 1.34iT - 17T^{2} \) |
| 19 | \( 1 + (-3.10 - 1.28i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-4.30 - 4.30i)T + 23iT^{2} \) |
| 29 | \( 1 + (0.600 - 1.44i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 + (4.03 - 1.67i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (5.34 + 5.34i)T + 41iT^{2} \) |
| 43 | \( 1 + (3.01 + 7.27i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 1.02iT - 47T^{2} \) |
| 53 | \( 1 + (-2.69 - 6.49i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-5.90 + 2.44i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (1.83 - 4.42i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (-2.58 + 6.23i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (5.75 - 5.75i)T - 71iT^{2} \) |
| 73 | \( 1 + (2.94 + 2.94i)T + 73iT^{2} \) |
| 79 | \( 1 + 15.4iT - 79T^{2} \) |
| 83 | \( 1 + (11.2 + 4.66i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (9.04 - 9.04i)T - 89iT^{2} \) |
| 97 | \( 1 - 8.41T + 97T^{2} \) |
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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
−9.823029995409943064548281529776, −8.976312487370220205868245865218, −8.664427028845385824342704310368, −7.24565203780761347237514004538, −6.41978029962938838087977833908, −5.66095871557260821377349837805, −5.07367703477058806133074416992, −3.30377040186003443929083055025, −2.32122337352128585915733188103, −1.37075379152253187755445832267,
1.19263057235737229629694382045, 2.83015726435964568509425364759, 3.51675512887393538436709512946, 4.63840294391634004695643863301, 5.86036410055171298145811497247, 6.70771095044501608686825464767, 7.02347736304945991657998351763, 8.552206467330714501078473990289, 9.579221682124988358787550229480, 9.944845527937342131833272935123