| L(s) = 1 | + 2·5-s − 6·11-s + 13-s + 17-s + 5·19-s − 6·23-s − 25-s − 6·29-s + 3·31-s − 3·37-s + 9·43-s + 4·47-s − 2·53-s − 12·55-s − 4·59-s + 2·61-s + 2·65-s + 15·67-s − 13·73-s + 5·79-s + 2·85-s + 16·89-s + 10·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.80·11-s + 0.277·13-s + 0.242·17-s + 1.14·19-s − 1.25·23-s − 1/5·25-s − 1.11·29-s + 0.538·31-s − 0.493·37-s + 1.37·43-s + 0.583·47-s − 0.274·53-s − 1.61·55-s − 0.520·59-s + 0.256·61-s + 0.248·65-s + 1.83·67-s − 1.52·73-s + 0.562·79-s + 0.216·85-s + 1.69·89-s + 1.02·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.82531566959342, −13.25945843666938, −13.05370676853427, −12.38927079768269, −11.93625490922130, −11.36671431141394, −10.75441811047199, −10.36916982466628, −9.985550035947748, −9.431249921519691, −9.102226500030351, −8.236124382622301, −7.864986647138964, −7.522875075189531, −6.861237527442557, −6.102363391476527, −5.678855246931657, −5.405046227957485, −4.824612928637446, −4.051235099640997, −3.455970402917523, −2.759737931563559, −2.272384223560277, −1.731344652245155, −0.8539124855527229, 0,
0.8539124855527229, 1.731344652245155, 2.272384223560277, 2.759737931563559, 3.455970402917523, 4.051235099640997, 4.824612928637446, 5.405046227957485, 5.678855246931657, 6.102363391476527, 6.861237527442557, 7.522875075189531, 7.864986647138964, 8.236124382622301, 9.102226500030351, 9.431249921519691, 9.985550035947748, 10.36916982466628, 10.75441811047199, 11.36671431141394, 11.93625490922130, 12.38927079768269, 13.05370676853427, 13.25945843666938, 13.82531566959342
