| L(s) = 1 | − 3-s + 4·7-s + 9-s − 6·13-s + 2·17-s − 4·21-s − 23-s − 5·25-s − 27-s − 4·31-s + 2·37-s + 6·39-s − 4·41-s − 4·43-s + 9·49-s − 2·51-s + 8·53-s + 4·59-s − 2·61-s + 4·63-s + 10·67-s + 69-s − 16·71-s + 10·73-s + 5·75-s + 16·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.66·13-s + 0.485·17-s − 0.872·21-s − 0.208·23-s − 25-s − 0.192·27-s − 0.718·31-s + 0.328·37-s + 0.960·39-s − 0.624·41-s − 0.609·43-s + 9/7·49-s − 0.280·51-s + 1.09·53-s + 0.520·59-s − 0.256·61-s + 0.503·63-s + 1.22·67-s + 0.120·69-s − 1.89·71-s + 1.17·73-s + 0.577·75-s + 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−15.06635502758331, −14.76371171749928, −14.40583770814578, −13.67473137606134, −13.24078794610103, −12.36515384612014, −11.98015802957792, −11.75197296784712, −10.99657255958241, −10.62835485365260, −9.864325278341199, −9.593238883187747, −8.744554558562465, −8.038315098463198, −7.711122847938367, −7.157199307477711, −6.535856376372076, −5.579327544508834, −5.313380356373070, −4.736000266886605, −4.179965287775946, −3.395694768488448, −2.287708404695402, −1.937844335706901, −1.001009250012805, 0,
1.001009250012805, 1.937844335706901, 2.287708404695402, 3.395694768488448, 4.179965287775946, 4.736000266886605, 5.313380356373070, 5.579327544508834, 6.535856376372076, 7.157199307477711, 7.711122847938367, 8.038315098463198, 8.744554558562465, 9.593238883187747, 9.864325278341199, 10.62835485365260, 10.99657255958241, 11.75197296784712, 11.98015802957792, 12.36515384612014, 13.24078794610103, 13.67473137606134, 14.40583770814578, 14.76371171749928, 15.06635502758331
