| L(s) = 1 | − 3-s − 5-s + 4·7-s − 2·9-s + 11-s + 13-s + 15-s + 17-s − 3·19-s − 4·21-s + 4·23-s + 25-s + 5·27-s − 3·29-s − 31-s − 33-s − 4·35-s − 4·37-s − 39-s + 6·43-s + 2·45-s − 3·47-s + 9·49-s − 51-s − 3·53-s − 55-s + 3·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s − 2/3·9-s + 0.301·11-s + 0.277·13-s + 0.258·15-s + 0.242·17-s − 0.688·19-s − 0.872·21-s + 0.834·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s − 0.179·31-s − 0.174·33-s − 0.676·35-s − 0.657·37-s − 0.160·39-s + 0.914·43-s + 0.298·45-s − 0.437·47-s + 9/7·49-s − 0.140·51-s − 0.412·53-s − 0.134·55-s + 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
| 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
−14.46298126599475, −14.31788585827913, −13.62312974896806, −12.95383613227732, −12.45542799996111, −11.81603399360343, −11.56530703102705, −11.01679984018963, −10.76041547598787, −10.20432090320194, −9.206211873223388, −8.877322343407335, −8.349888809820238, −7.832439779529188, −7.355652666497558, −6.667917482209454, −6.065152498337256, −5.498334650312946, −4.948646294619809, −4.536886712144361, −3.833827644409356, −3.160885605541603, −2.351693988991592, −1.601858800655579, −0.9455672549411977, 0,
0.9455672549411977, 1.601858800655579, 2.351693988991592, 3.160885605541603, 3.833827644409356, 4.536886712144361, 4.948646294619809, 5.498334650312946, 6.065152498337256, 6.667917482209454, 7.355652666497558, 7.832439779529188, 8.349888809820238, 8.877322343407335, 9.206211873223388, 10.20432090320194, 10.76041547598787, 11.01679984018963, 11.56530703102705, 11.81603399360343, 12.45542799996111, 12.95383613227732, 13.62312974896806, 14.31788585827913, 14.46298126599475
