| L(s) = 1 | − 3-s + 4·7-s + 9-s − 2·13-s + 4·17-s + 6·19-s − 4·21-s − 6·23-s − 27-s − 10·37-s + 2·39-s − 10·41-s − 43-s + 6·47-s + 9·49-s − 4·51-s + 2·53-s − 6·57-s − 4·59-s − 8·61-s + 4·63-s + 12·67-s + 6·69-s − 8·71-s − 6·73-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.970·17-s + 1.37·19-s − 0.872·21-s − 1.25·23-s − 0.192·27-s − 1.64·37-s + 0.320·39-s − 1.56·41-s − 0.152·43-s + 0.875·47-s + 9/7·49-s − 0.560·51-s + 0.274·53-s − 0.794·57-s − 0.520·59-s − 1.02·61-s + 0.503·63-s + 1.46·67-s + 0.722·69-s − 0.949·71-s − 0.702·73-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.66072631979302, −14.19955841972896, −13.83806955510969, −13.35625691111366, −12.36611639686585, −12.05101595645269, −11.80444267199471, −11.28518917827159, −10.44723074348296, −10.35314501712787, −9.639188402788928, −9.041491960553348, −8.287650053289337, −7.925388811498126, −7.362782328004728, −6.963525618540505, −6.032979349709262, −5.540681195765330, −5.019240966245308, −4.711148865860127, −3.807105751101989, −3.290125809730345, −2.277650920225202, −1.631760897559638, −1.077032339433796, 0,
1.077032339433796, 1.631760897559638, 2.277650920225202, 3.290125809730345, 3.807105751101989, 4.711148865860127, 5.019240966245308, 5.540681195765330, 6.032979349709262, 6.963525618540505, 7.362782328004728, 7.925388811498126, 8.287650053289337, 9.041491960553348, 9.639188402788928, 10.35314501712787, 10.44723074348296, 11.28518917827159, 11.80444267199471, 12.05101595645269, 12.36611639686585, 13.35625691111366, 13.83806955510969, 14.19955841972896, 14.66072631979302
