| L(s) = 1 | + (0.460 − 1.94i)2-s + (−3.57 − 1.79i)4-s + (8.07 − 4.66i)5-s + (4.91 + 2.83i)7-s + (−5.13 + 6.13i)8-s + (−5.35 − 17.8i)10-s + (1.85 − 3.21i)11-s + (11.0 − 6.40i)13-s + (7.78 − 8.25i)14-s + (9.56 + 12.8i)16-s − 11.1·17-s − 13.1·19-s + (−37.2 + 2.18i)20-s + (−5.39 − 5.09i)22-s + (−20.2 + 11.6i)23-s + ⋯ |
| L(s) = 1 | + (0.230 − 0.973i)2-s + (−0.893 − 0.448i)4-s + (1.61 − 0.932i)5-s + (0.701 + 0.405i)7-s + (−0.642 + 0.766i)8-s + (−0.535 − 1.78i)10-s + (0.168 − 0.292i)11-s + (0.853 − 0.492i)13-s + (0.555 − 0.589i)14-s + (0.597 + 0.801i)16-s − 0.657·17-s − 0.692·19-s + (−1.86 + 0.109i)20-s + (−0.245 − 0.231i)22-s + (−0.878 + 0.507i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.24893 - 1.69891i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24893 - 1.69891i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.460 + 1.94i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-8.07 + 4.66i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-4.91 - 2.83i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-1.85 + 3.21i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-11.0 + 6.40i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 11.1T + 289T^{2} \) |
| 19 | \( 1 + 13.1T + 361T^{2} \) |
| 23 | \( 1 + (20.2 - 11.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (14.6 + 8.47i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (3.32 - 1.91i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 13.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (3.05 + 5.29i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (11.3 - 19.7i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-49.8 - 28.8i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 60.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-55.3 - 95.8i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-73.2 - 42.3i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-16.0 - 27.8i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 38.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 13.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + (4.14 + 2.39i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-2.70 + 4.68i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 98.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-42.1 + 73.0i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−11.84451688922742824762809881110, −10.81929016064450436004878029381, −9.931799027116051543456863668743, −8.954826335443679530813237179542, −8.403316463872672649878086913499, −6.05302821197915015554075721629, −5.44897991258002665454496827610, −4.25596405179553510505555345877, −2.35894510307043343565869016264, −1.30811167084077401670713319110,
2.03795997020457917268610272928, 3.92877590191384603361665719730, 5.30725193178266543678373544576, 6.36160412823680424796685627519, 6.94639218831700998162448074655, 8.346936839097824969899902728310, 9.353379074127032403663395065043, 10.30776183568941013422506427131, 11.24851378721764352211102657417, 12.83449504466697298861407243203
