| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s + 2·13-s + 2·15-s − 6·17-s + 8·19-s + 21-s − 4·23-s − 25-s − 27-s + 2·29-s − 8·31-s + 2·35-s − 6·37-s − 2·39-s − 6·41-s − 8·43-s − 2·45-s − 4·47-s + 49-s + 6·51-s − 10·53-s − 8·57-s + 4·59-s − 14·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s − 1.45·17-s + 1.83·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.338·35-s − 0.986·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.298·45-s − 0.583·47-s + 1/7·49-s + 0.840·51-s − 1.37·53-s − 1.05·57-s + 0.520·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.41467210703812, −13.03139035122112, −12.44275477262970, −11.96242432494170, −11.65897073588439, −11.06610444839143, −10.96793950441693, −10.03334315787028, −9.870706996612386, −9.146910893275135, −8.714940850382708, −8.143007480438065, −7.664332135434100, −7.059470950902542, −6.820075699702826, −6.067515459432655, −5.740889346953981, −4.910773730159927, −4.696869778270438, −3.898522192563998, −3.410713860646533, −3.120937056557855, −1.950610854052159, −1.610225616118855, −0.5651516645269264, 0,
0.5651516645269264, 1.610225616118855, 1.950610854052159, 3.120937056557855, 3.410713860646533, 3.898522192563998, 4.696869778270438, 4.910773730159927, 5.740889346953981, 6.067515459432655, 6.820075699702826, 7.059470950902542, 7.664332135434100, 8.143007480438065, 8.714940850382708, 9.146910893275135, 9.870706996612386, 10.03334315787028, 10.96793950441693, 11.06610444839143, 11.65897073588439, 11.96242432494170, 12.44275477262970, 13.03139035122112, 13.41467210703812
