| L(s) = 1 | − 5.56·3-s − 3.05·5-s + 21.9·9-s − 0.122i·11-s − 4.11·13-s + 17.0·15-s − 20.6i·17-s − 8.93·19-s + 15.0·23-s − 15.6·25-s − 71.8·27-s + 31.6i·29-s + 26.5i·31-s + 0.680i·33-s + 29.0i·37-s + ⋯ |
| L(s) = 1 | − 1.85·3-s − 0.611·5-s + 2.43·9-s − 0.0111i·11-s − 0.316·13-s + 1.13·15-s − 1.21i·17-s − 0.470·19-s + 0.653·23-s − 0.625·25-s − 2.65·27-s + 1.09i·29-s + 0.857i·31-s + 0.0206i·33-s + 0.784i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4457646501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4457646501\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 5.56T + 9T^{2} \) |
| 5 | \( 1 + 3.05T + 25T^{2} \) |
| 11 | \( 1 + 0.122iT - 121T^{2} \) |
| 13 | \( 1 + 4.11T + 169T^{2} \) |
| 17 | \( 1 + 20.6iT - 289T^{2} \) |
| 19 | \( 1 + 8.93T + 361T^{2} \) |
| 23 | \( 1 - 15.0T + 529T^{2} \) |
| 29 | \( 1 - 31.6iT - 841T^{2} \) |
| 31 | \( 1 - 26.5iT - 961T^{2} \) |
| 37 | \( 1 - 29.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 9.26iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 45.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 79.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 63.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 28.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 25.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 75.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 2.81T + 5.04e3T^{2} \) |
| 73 | \( 1 + 12.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 71.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 30.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 17.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 26.1iT - 9.40e3T^{2} \) |
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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
−9.334763415497292625518786604286, −8.105687784527535097904152916853, −7.17071051940272606879782845992, −6.70485402096399451715558800135, −5.76331132195682497458950126805, −4.91847955799555565077283303604, −4.44947907046727546264549588451, −3.12774775137991792545397909417, −1.41287970192576799667212382893, −0.27956951996772544803845843242,
0.65535460021903698889698852816, 2.02102709341646817015241199585, 3.88290343972514913356323757235, 4.38257035745542325196307087829, 5.49162376198947323469802304204, 5.98253689900279976314248408362, 6.89454257199378911751150330957, 7.54707187183646227790612486120, 8.519323097271619358945245351355, 9.687957391888669646907808829179
