| L(s) = 1 | − 4·7-s + 4·11-s + 13-s + 6·17-s − 4·23-s − 6·29-s − 8·31-s − 2·37-s − 10·41-s − 4·43-s + 8·47-s + 9·49-s + 2·53-s + 4·59-s − 14·61-s − 12·67-s + 8·71-s + 10·73-s − 16·77-s + 4·83-s − 10·89-s − 4·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.20·11-s + 0.277·13-s + 1.45·17-s − 0.834·23-s − 1.11·29-s − 1.43·31-s − 0.328·37-s − 1.56·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s + 0.274·53-s + 0.520·59-s − 1.79·61-s − 1.46·67-s + 0.949·71-s + 1.17·73-s − 1.82·77-s + 0.439·83-s − 1.05·89-s − 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.45069526877810, −12.70084508773103, −12.51755053202957, −11.98074307187741, −11.64847612478263, −10.99252666974400, −10.39631157814462, −10.01283382217243, −9.590389857653557, −9.118370779391470, −8.802084126952917, −8.111654844450653, −7.356659510681164, −7.249110903023807, −6.420392867261061, −6.210628739412007, −5.616855006285084, −5.207972794881043, −4.284342480722178, −3.709190711278421, −3.468250527616159, −3.029929399791281, −2.013643354431423, −1.586688817866588, −0.7331667545298475, 0,
0.7331667545298475, 1.586688817866588, 2.013643354431423, 3.029929399791281, 3.468250527616159, 3.709190711278421, 4.284342480722178, 5.207972794881043, 5.616855006285084, 6.210628739412007, 6.420392867261061, 7.249110903023807, 7.356659510681164, 8.111654844450653, 8.802084126952917, 9.118370779391470, 9.590389857653557, 10.01283382217243, 10.39631157814462, 10.99252666974400, 11.64847612478263, 11.98074307187741, 12.51755053202957, 12.70084508773103, 13.45069526877810
