L(s) = 1 | + (−0.291 + 0.504i)3-s − 1.68i·5-s + (−0.866 + 0.5i)7-s + (1.33 + 2.30i)9-s + (0.315 + 0.182i)11-s + (1.80 + 3.12i)13-s + (0.851 + 0.491i)15-s + (−1.59 − 2.75i)17-s + (−1.25 + 0.721i)19-s − 0.582i·21-s + (2.54 − 4.40i)23-s + 2.15·25-s − 3.29·27-s + (−4.09 + 7.09i)29-s + 4.69i·31-s + ⋯ |
L(s) = 1 | + (−0.168 + 0.291i)3-s − 0.754i·5-s + (−0.327 + 0.188i)7-s + (0.443 + 0.768i)9-s + (0.0952 + 0.0549i)11-s + (0.499 + 0.866i)13-s + (0.219 + 0.126i)15-s + (−0.386 − 0.669i)17-s + (−0.286 + 0.165i)19-s − 0.127i·21-s + (0.529 − 0.917i)23-s + 0.430·25-s − 0.634·27-s + (−0.761 + 1.31i)29-s + 0.843i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.443840800\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.443840800\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (-1.80 - 3.12i)T \) |
good | 3 | \( 1 + (0.291 - 0.504i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + 1.68iT - 5T^{2} \) |
| 11 | \( 1 + (-0.315 - 0.182i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.59 + 2.75i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.25 - 0.721i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.54 + 4.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.09 - 7.09i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4.69iT - 31T^{2} \) |
| 37 | \( 1 + (-5.46 - 3.15i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.04 + 2.91i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.386 - 0.669i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 12.7iT - 47T^{2} \) |
| 53 | \( 1 - 1.37T + 53T^{2} \) |
| 59 | \( 1 + (-8.10 + 4.68i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.51 - 7.81i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.6 - 6.73i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.13 + 3.54i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 2.16iT - 73T^{2} \) |
| 79 | \( 1 - 6.88T + 79T^{2} \) |
| 83 | \( 1 + 0.567iT - 83T^{2} \) |
| 89 | \( 1 + (0.986 + 0.569i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.86 + 3.96i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.494446070798544718406479490204, −8.937234748899777243585660805185, −8.238765913703004144699063899888, −7.10960095326458898879432446482, −6.48031432367857031904880653207, −5.24505808283747081288006648100, −4.74139342650271068151717387015, −3.82531102963144305087402869742, −2.48381447161580878849033899631, −1.24453201947198164410251868041,
0.66567053017726681481662773492, 2.16299073063294698636216958086, 3.41910141972432898384396565164, 3.99639622580362013997099246264, 5.44049996788862542773304773497, 6.27155973516686243593449189049, 6.84173600769625610259136862184, 7.65540738885515502181018402886, 8.527323834645394598509367718827, 9.552702270136097936375980756012