| L(s) = 1 | + 0.732·3-s + 0.267·5-s − 0.732·7-s − 2.46·9-s + 2.46·13-s + 0.196·15-s + 5.73·17-s − 6.73·19-s − 0.535·21-s + 8.19·23-s − 4.92·25-s − 4·27-s − 2.46·29-s − 1.26·31-s − 0.196·35-s − 2.26·37-s + 1.80·39-s + 4.26·41-s − 8·43-s − 0.660·45-s − 6.73·47-s − 6.46·49-s + 4.19·51-s + 9.19·53-s − 4.92·57-s − 6.53·59-s − 13.4·61-s + ⋯ |
| L(s) = 1 | + 0.422·3-s + 0.119·5-s − 0.276·7-s − 0.821·9-s + 0.683·13-s + 0.0506·15-s + 1.39·17-s − 1.54·19-s − 0.116·21-s + 1.70·23-s − 0.985·25-s − 0.769·27-s − 0.457·29-s − 0.227·31-s − 0.0331·35-s − 0.372·37-s + 0.288·39-s + 0.666·41-s − 1.21·43-s − 0.0984·45-s − 0.981·47-s − 0.923·49-s + 0.587·51-s + 1.26·53-s − 0.652·57-s − 0.850·59-s − 1.72·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 - 0.267T + 5T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 - 5.73T + 17T^{2} \) |
| 19 | \( 1 + 6.73T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 + 2.26T + 37T^{2} \) |
| 41 | \( 1 - 4.26T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 6.73T + 47T^{2} \) |
| 53 | \( 1 - 9.19T + 53T^{2} \) |
| 59 | \( 1 + 6.53T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 - 4.92T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 + 9.53T + 89T^{2} \) |
| 97 | \( 1 - 1.92T + 97T^{2} \) |
| 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
−7.71599810918461130844422599878, −6.73973497440728540464030149864, −6.13655358245100888352837171318, −5.49885275588768465405204704459, −4.72450140538941229266572354451, −3.62424411642755692526507802601, −3.25784957647993034050141955457, −2.30562841074614672636499545457, −1.35382375448858145518966389937, 0,
1.35382375448858145518966389937, 2.30562841074614672636499545457, 3.25784957647993034050141955457, 3.62424411642755692526507802601, 4.72450140538941229266572354451, 5.49885275588768465405204704459, 6.13655358245100888352837171318, 6.73973497440728540464030149864, 7.71599810918461130844422599878
