| L(s) = 1 | − 5.65i·2-s − 32.0·4-s − 110. i·5-s + 326.·7-s + 181. i·8-s − 626.·10-s − 625. i·11-s − 2.02e3·13-s − 1.84e3i·14-s + 1.02e3·16-s − 4.12e3i·17-s + 1.21e4·19-s + 3.54e3i·20-s − 3.53e3·22-s − 2.14e4i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.885i·5-s + 0.951·7-s + 0.353i·8-s − 0.626·10-s − 0.470i·11-s − 0.923·13-s − 0.672i·14-s + 0.250·16-s − 0.839i·17-s + 1.77·19-s + 0.442i·20-s − 0.332·22-s − 1.76i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.472906966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472906966\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5.65iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 110. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 326.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 625. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.02e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 4.12e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.21e4T + 4.70e7T^{2} \) |
| 23 | \( 1 + 2.14e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.47e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 4.05e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.00e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 5.92e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 8.79e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.29e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.65e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 5.32e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.66e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.07e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 1.86e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 2.42e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 1.26e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 6.89e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 4.13e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 9.78e5T + 8.32e11T^{2} \) |
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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
−11.43931097440848842179517262966, −10.30383826747226196557663939531, −9.196081630002515322763283421803, −8.398997152021743380889055494441, −7.22634204834298859785296830750, −5.24901292799266953203470567938, −4.75386448346962458356244972214, −3.08819265673859249241345467081, −1.59575459026104394912794821570, −0.43685981030268812081377515006,
1.64968759588892319160617253438, 3.33990678986339634890996608692, 4.81934590504708385098520478609, 5.83674871636874586217008859408, 7.35464763393777658979514575014, 7.62735640825490909454941090713, 9.177044171455404847372426014928, 10.14043141109318585100238098880, 11.25920613367504847194209501960, 12.17951607101778170540382843210
