| L(s) = 1 | − 1.11·3-s + 1.47·5-s − 7-s − 1.75·9-s + 2·11-s − 0.715·13-s − 1.64·15-s − 17-s + 1.28·19-s + 1.11·21-s + 2.22·23-s − 2.83·25-s + 5.30·27-s + 9.17·29-s − 6.75·31-s − 2.22·33-s − 1.47·35-s − 8.45·37-s + 0.798·39-s − 6.77·41-s + 4.39·43-s − 2.58·45-s + 2.22·47-s + 49-s + 1.11·51-s − 6.64·53-s + 2.94·55-s + ⋯ |
| L(s) = 1 | − 0.643·3-s + 0.658·5-s − 0.377·7-s − 0.585·9-s + 0.603·11-s − 0.198·13-s − 0.423·15-s − 0.242·17-s + 0.294·19-s + 0.243·21-s + 0.464·23-s − 0.566·25-s + 1.02·27-s + 1.70·29-s − 1.21·31-s − 0.388·33-s − 0.248·35-s − 1.39·37-s + 0.127·39-s − 1.05·41-s + 0.670·43-s − 0.385·45-s + 0.325·47-s + 0.142·49-s + 0.156·51-s − 0.913·53-s + 0.397·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.11T + 3T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 0.715T + 13T^{2} \) |
| 19 | \( 1 - 1.28T + 19T^{2} \) |
| 23 | \( 1 - 2.22T + 23T^{2} \) |
| 29 | \( 1 - 9.17T + 29T^{2} \) |
| 31 | \( 1 + 6.75T + 31T^{2} \) |
| 37 | \( 1 + 8.45T + 37T^{2} \) |
| 41 | \( 1 + 6.77T + 41T^{2} \) |
| 43 | \( 1 - 4.39T + 43T^{2} \) |
| 47 | \( 1 - 2.22T + 47T^{2} \) |
| 53 | \( 1 + 6.64T + 53T^{2} \) |
| 59 | \( 1 - 5.28T + 59T^{2} \) |
| 61 | \( 1 + 0.885T + 61T^{2} \) |
| 67 | \( 1 - 4.90T + 67T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 2.37T + 83T^{2} \) |
| 89 | \( 1 - 7.51T + 89T^{2} \) |
| 97 | \( 1 - 5.85T + 97T^{2} \) |
| 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
−7.36078431838473837543093332180, −6.59018045463401866969382811145, −6.25494735276514276383087120823, −5.37827004198521712058872627408, −4.99397693756462138593625798734, −3.90863622904328578692077003490, −3.10489677150231877786290367268, −2.22686654450316719803388932934, −1.19714557101758753292908131097, 0,
1.19714557101758753292908131097, 2.22686654450316719803388932934, 3.10489677150231877786290367268, 3.90863622904328578692077003490, 4.99397693756462138593625798734, 5.37827004198521712058872627408, 6.25494735276514276383087120823, 6.59018045463401866969382811145, 7.36078431838473837543093332180
