| L(s) = 1 | + 2.02i·2-s − 2.09·4-s − 0.182·5-s − 0.197i·8-s − 0.369i·10-s − 3.75i·11-s + 5.85i·13-s − 3.79·16-s + 4.31·17-s + 3.82i·19-s + 0.383·20-s + 7.60·22-s + 4.66i·23-s − 4.96·25-s − 11.8·26-s + ⋯ |
| L(s) = 1 | + 1.43i·2-s − 1.04·4-s − 0.0817·5-s − 0.0699i·8-s − 0.116i·10-s − 1.13i·11-s + 1.62i·13-s − 0.948·16-s + 1.04·17-s + 0.877i·19-s + 0.0857·20-s + 1.62·22-s + 0.972i·23-s − 0.993·25-s − 2.32·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.156291168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.156291168\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.02iT - 2T^{2} \) |
| 5 | \( 1 + 0.182T + 5T^{2} \) |
| 11 | \( 1 + 3.75iT - 11T^{2} \) |
| 13 | \( 1 - 5.85iT - 13T^{2} \) |
| 17 | \( 1 - 4.31T + 17T^{2} \) |
| 19 | \( 1 - 3.82iT - 19T^{2} \) |
| 23 | \( 1 - 4.66iT - 23T^{2} \) |
| 29 | \( 1 + 5.70iT - 29T^{2} \) |
| 31 | \( 1 - 8.38iT - 31T^{2} \) |
| 37 | \( 1 + 8.44T + 37T^{2} \) |
| 41 | \( 1 - 1.51T + 41T^{2} \) |
| 43 | \( 1 + 3.00T + 43T^{2} \) |
| 47 | \( 1 - 0.591T + 47T^{2} \) |
| 53 | \( 1 - 13.0iT - 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 1.40iT - 61T^{2} \) |
| 67 | \( 1 - 7.78T + 67T^{2} \) |
| 71 | \( 1 + 5.78iT - 71T^{2} \) |
| 73 | \( 1 - 4.53iT - 73T^{2} \) |
| 79 | \( 1 + 7.88T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 - 2.33iT - 97T^{2} \) |
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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
−9.760757632328674466885342940480, −9.018593025230161093909957010419, −8.269212892605955936077961025793, −7.60885279735286109386865221326, −6.80437983655947433240056306712, −5.98541561260412737378903795972, −5.44917876230984421016958931940, −4.31461399027911171888649018403, −3.34710302753998051724906069562, −1.68138109653485098126799636450,
0.47281457922118272368932415327, 1.83237877243151312001348497273, 2.83217138078456155672310686695, 3.64650606513315842464950025647, 4.67931165644039315366650519384, 5.53840596103904323203637302747, 6.81249148419863776879439313910, 7.65875415007989409950589418040, 8.546406120839070626239330307410, 9.627471790564901177283941977022
