| 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 | \( 1 - 6.35e5T + 8.32e11T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.85165338453345330118070099469, −10.73601906856893470645257434051, −9.502243435358710679696469732803, −8.469843313200101659875849527252, −7.63546001098204671774757815669, −6.54788152774713377626116249810, −4.90610818941252806169335209844, −4.59817429874603624087006439571, −2.33956228264647617406167529844, −0.71950281290307236442619609934,
1.14952042000390547945415262850, 2.33825198967412763132903952697, 3.84424751980444640369682267344, 4.98315055475279062117779544222, 6.30281502183844313630808831561, 7.965355468178087498338313004863, 8.541698070128694006380197603504, 9.995337576224243255424879686258, 10.97130532966640414313863385353, 11.46008662905605465657793688630
