L(s) = 1 | + (0.665 + 1.24i)2-s + (3.13 − 0.786i)3-s + (−1.11 + 1.66i)4-s + (−1.08 − 0.511i)5-s + (3.06 + 3.39i)6-s + (−0.258 − 0.851i)7-s + (−2.81 − 0.286i)8-s + (6.59 − 3.52i)9-s + (−0.0811 − 1.69i)10-s + (1.48 − 0.219i)11-s + (−2.19 + 6.08i)12-s + (−2.17 + 6.07i)13-s + (0.891 − 0.889i)14-s + (−3.80 − 0.756i)15-s + (−1.51 − 3.70i)16-s + (−2.51 + 0.500i)17-s + ⋯ |
L(s) = 1 | + (0.470 + 0.882i)2-s + (1.81 − 0.454i)3-s + (−0.557 + 0.830i)4-s + (−0.484 − 0.228i)5-s + (1.25 + 1.38i)6-s + (−0.0976 − 0.321i)7-s + (−0.994 − 0.101i)8-s + (2.19 − 1.17i)9-s + (−0.0256 − 0.534i)10-s + (0.446 − 0.0663i)11-s + (−0.633 + 1.75i)12-s + (−0.602 + 1.68i)13-s + (0.238 − 0.237i)14-s + (−0.981 − 0.195i)15-s + (−0.378 − 0.925i)16-s + (−0.609 + 0.121i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.10908 + 0.892782i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.10908 + 0.892782i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.665 - 1.24i)T \) |
good | 3 | \( 1 + (-3.13 + 0.786i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (1.08 + 0.511i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (0.258 + 0.851i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (-1.48 + 0.219i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (2.17 - 6.07i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (2.51 - 0.500i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (3.15 + 2.85i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-2.00 + 0.197i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (-0.298 - 0.403i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (5.73 + 2.37i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-0.329 + 6.71i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (6.33 - 7.71i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (1.69 - 6.75i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (-7.65 - 5.11i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (-1.28 + 1.73i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (-0.666 - 1.86i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (-5.59 + 9.32i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (-13.3 - 7.98i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (3.21 - 6.00i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-0.779 + 2.56i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (4.86 + 7.27i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-0.695 - 14.1i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (1.00 + 0.0992i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (-0.850 + 2.05i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−12.63275053712187739159807469458, −11.53399489737870960396979383841, −9.559735118806380307860435695529, −8.960036726798019474803718385065, −8.195859785560664643462680346038, −7.17095194535064398769756247727, −6.65030923076538109879704482631, −4.43639949179189873356772663606, −3.84753249511263700074063496763, −2.32816302743219020812605360256,
2.15579236877016340699778621931, 3.23318526768576208330561383895, 3.97787827210617178005374963511, 5.30809256855171681022167319548, 7.20391682298231883692640332749, 8.365342555968935549425715828587, 9.043093881863667932490815339963, 10.07318319661818482123327978188, 10.67476578527517273404142018248, 12.10731123687482818772889381635