| L(s) = 1 | − 0.478·2-s − 1.77·4-s + 5-s − 0.0239·7-s + 1.80·8-s − 0.478·10-s − 2.56·11-s + 0.476·13-s + 0.0114·14-s + 2.67·16-s − 2.22·17-s − 4.21·19-s − 1.77·20-s + 1.22·22-s + 7.85·23-s + 25-s − 0.228·26-s + 0.0423·28-s − 2.13·29-s + 2.50·31-s − 4.89·32-s + 1.06·34-s − 0.0239·35-s + 6.23·37-s + 2.01·38-s + 1.80·40-s − 5.58·41-s + ⋯ |
| L(s) = 1 | − 0.338·2-s − 0.885·4-s + 0.447·5-s − 0.00904·7-s + 0.638·8-s − 0.151·10-s − 0.774·11-s + 0.132·13-s + 0.00306·14-s + 0.669·16-s − 0.540·17-s − 0.967·19-s − 0.395·20-s + 0.262·22-s + 1.63·23-s + 0.200·25-s − 0.0447·26-s + 0.00801·28-s − 0.395·29-s + 0.449·31-s − 0.864·32-s + 0.182·34-s − 0.00404·35-s + 1.02·37-s + 0.327·38-s + 0.285·40-s − 0.871·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 + 0.478T + 2T^{2} \) |
| 7 | \( 1 + 0.0239T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 0.476T + 13T^{2} \) |
| 17 | \( 1 + 2.22T + 17T^{2} \) |
| 19 | \( 1 + 4.21T + 19T^{2} \) |
| 23 | \( 1 - 7.85T + 23T^{2} \) |
| 29 | \( 1 + 2.13T + 29T^{2} \) |
| 31 | \( 1 - 2.50T + 31T^{2} \) |
| 37 | \( 1 - 6.23T + 37T^{2} \) |
| 41 | \( 1 + 5.58T + 41T^{2} \) |
| 43 | \( 1 + 0.503T + 43T^{2} \) |
| 47 | \( 1 - 5.72T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 8.99T + 59T^{2} \) |
| 61 | \( 1 + 2.20T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 3.83T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 7.47T + 83T^{2} \) |
| 97 | \( 1 + 5.75T + 97T^{2} \) |
| 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
−8.262069061807867146607555888679, −7.48544435018936113637600014663, −6.63655417028368950803113316452, −5.81197218033885580181973421765, −4.95014889912437157798712643471, −4.48787903223739485019419976409, −3.38741965990747467321349104146, −2.42498529533534314368386740552, −1.26783167506854778239177149219, 0,
1.26783167506854778239177149219, 2.42498529533534314368386740552, 3.38741965990747467321349104146, 4.48787903223739485019419976409, 4.95014889912437157798712643471, 5.81197218033885580181973421765, 6.63655417028368950803113316452, 7.48544435018936113637600014663, 8.262069061807867146607555888679
