| L(s) = 1 | − 3.65i·2-s − 9.32·4-s − 2.23i·5-s + 7.16·7-s + 19.4i·8-s − 8.16·10-s − 5.42i·11-s + 9.81·13-s − 26.1i·14-s + 33.6·16-s + 12.2i·17-s + 6.32·19-s + 20.8i·20-s − 19.8·22-s + 12.0i·23-s + ⋯ |
| L(s) = 1 | − 1.82i·2-s − 2.33·4-s − 0.447i·5-s + 1.02·7-s + 2.42i·8-s − 0.816·10-s − 0.493i·11-s + 0.754·13-s − 1.86i·14-s + 2.10·16-s + 0.719i·17-s + 0.332·19-s + 1.04i·20-s − 0.900·22-s + 0.523i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.316063 - 0.994417i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.316063 - 0.994417i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| good | 2 | \( 1 + 3.65iT - 4T^{2} \) |
| 7 | \( 1 - 7.16T + 49T^{2} \) |
| 11 | \( 1 + 5.42iT - 121T^{2} \) |
| 13 | \( 1 - 9.81T + 169T^{2} \) |
| 17 | \( 1 - 12.2iT - 289T^{2} \) |
| 19 | \( 1 - 6.32T + 361T^{2} \) |
| 23 | \( 1 - 12.0iT - 529T^{2} \) |
| 29 | \( 1 - 44.9iT - 841T^{2} \) |
| 31 | \( 1 + 58.2T + 961T^{2} \) |
| 37 | \( 1 - 66.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 16.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 43.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 40.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 13.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 25.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 35.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 26.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 92.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 60.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 96.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 79.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 1.07T + 9.40e3T^{2} \) |
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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
−14.69340882430047786698746792151, −13.55812231158382929072811776693, −12.61414924987039423003898016838, −11.40597117304721003355288277868, −10.76183130037221657123865468948, −9.254184688971605489538064157133, −8.231193285766477630363686792534, −5.23203266708047080026597361022, −3.69445399698654993586410750022, −1.53075393037827689856484300009,
4.48094676220591938456885133232, 5.88826827530437056144438716215, 7.26304874367848277594444184612, 8.206313514490785855644879291545, 9.555624276209101887261676726690, 11.32129944365197937474432785359, 13.18217674315845841497321198002, 14.27796181601394673546581063846, 14.94683057931806526875849824634, 15.96631794951471653631147116858
