| L(s) = 1 | + (−1.23 − 0.509i)2-s + (−0.523 − 2.63i)3-s + (−1.57 − 1.57i)4-s + (1.35 + 0.902i)5-s + (−0.696 + 3.50i)6-s + (7.37 − 4.92i)7-s + (3.17 + 7.66i)8-s + (1.66 − 0.690i)9-s + (−1.20 − 1.79i)10-s + (8.12 + 1.61i)11-s + (−3.31 + 4.96i)12-s + (16.1 − 16.1i)13-s + (−11.5 + 2.30i)14-s + (1.66 − 4.02i)15-s − 2.13i·16-s + ⋯ |
| L(s) = 1 | + (−0.615 − 0.254i)2-s + (−0.174 − 0.876i)3-s + (−0.393 − 0.393i)4-s + (0.270 + 0.180i)5-s + (−0.116 + 0.584i)6-s + (1.05 − 0.703i)7-s + (0.396 + 0.957i)8-s + (0.185 − 0.0767i)9-s + (−0.120 − 0.179i)10-s + (0.738 + 0.146i)11-s + (−0.276 + 0.413i)12-s + (1.24 − 1.24i)13-s + (−0.827 + 0.164i)14-s + (0.111 − 0.268i)15-s − 0.133i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.636075 - 1.08549i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.636075 - 1.08549i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (1.23 + 0.509i)T + (2.82 + 2.82i)T^{2} \) |
| 3 | \( 1 + (0.523 + 2.63i)T + (-8.31 + 3.44i)T^{2} \) |
| 5 | \( 1 + (-1.35 - 0.902i)T + (9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (-7.37 + 4.92i)T + (18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (-8.12 - 1.61i)T + (111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (-16.1 + 16.1i)T - 169iT^{2} \) |
| 19 | \( 1 + (-1.20 - 0.500i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (5.21 - 26.2i)T + (-488. - 202. i)T^{2} \) |
| 29 | \( 1 + (9.19 - 13.7i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 + (13.4 - 2.68i)T + (887. - 367. i)T^{2} \) |
| 37 | \( 1 + (-3.77 - 18.9i)T + (-1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (21.9 - 14.6i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-21.8 + 9.06i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-26.8 + 26.8i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (26.1 + 10.8i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (9.12 + 22.0i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-28.6 - 42.9i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 + 70.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (20.4 + 102. i)T + (-4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-29.5 - 19.7i)T + (2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-7.11 - 1.41i)T + (5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (51.5 - 124. i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-38.6 - 38.6i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (44.0 - 65.9i)T + (-3.60e3 - 8.69e3i)T^{2} \) |
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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
−11.08635699996785400278324427837, −10.43559671062214058789476570246, −9.434890159049910691823992675419, −8.260022723251158587615244388860, −7.63773360504026605896455027751, −6.38103288947896910402055904912, −5.32120254137104370168583555134, −3.91895291443584601599637735121, −1.70995011386412585181517248611, −0.949677871245320882482966949689,
1.60503192196202363952864524997, 3.88344023011251486619246031103, 4.56515505751408064356213031390, 5.86362950184916193754219301744, 7.18733622899143109644611768675, 8.458013219269524648262708390237, 8.993620069197252989508562581785, 9.717666108545483728509705128960, 10.92995203481189768930348861358, 11.62010702431533077137369490951
