L(s) = 1 | − 5.65i·2-s − 32.0·4-s + 45.6i·5-s + 490.·7-s + 181. i·8-s + 258.·10-s − 1.00e3i·11-s − 933.·13-s − 2.77e3i·14-s + 1.02e3·16-s + 8.09e3i·17-s − 7.72e3·19-s − 1.46e3i·20-s − 5.70e3·22-s + 1.36e4i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.365i·5-s + 1.42·7-s + 0.353i·8-s + 0.258·10-s − 0.757i·11-s − 0.424·13-s − 1.01i·14-s + 0.250·16-s + 1.64i·17-s − 1.12·19-s − 0.182i·20-s − 0.535·22-s + 1.12i·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
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.083053034\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.083053034\) |
\(L(4)\) |
|
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 | \( 1 - 45.6iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 490.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.00e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 933.T + 4.82e6T^{2} \) |
| 17 | \( 1 - 8.09e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 7.72e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.36e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.26e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.41e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 9.20e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 3.58e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.91e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.52e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.36e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.56e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 3.98e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 3.20e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.04e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.93e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 8.98e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.78e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 8.26e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 6.35e5T + 8.32e11T^{2} \) |
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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
−11.46008662905605465657793688630, −10.97130532966640414313863385353, −9.995337576224243255424879686258, −8.541698070128694006380197603504, −7.965355468178087498338313004863, −6.30281502183844313630808831561, −4.98315055475279062117779544222, −3.84424751980444640369682267344, −2.33825198967412763132903952697, −1.14952042000390547945415262850,
0.71950281290307236442619609934, 2.33956228264647617406167529844, 4.59817429874603624087006439571, 4.90610818941252806169335209844, 6.54788152774713377626116249810, 7.63546001098204671774757815669, 8.469843313200101659875849527252, 9.502243435358710679696469732803, 10.73601906856893470645257434051, 11.85165338453345330118070099469