| L(s) = 1 | + 0.311·2-s + 2.21·3-s − 1.90·4-s + 5-s + 0.688·6-s − 1.59·7-s − 1.21·8-s + 1.90·9-s + 0.311·10-s + 1.31·11-s − 4.21·12-s + 3.52·13-s − 0.495·14-s + 2.21·15-s + 3.42·16-s + 0.592·18-s + 4.42·19-s − 1.90·20-s − 3.52·21-s + 0.407·22-s + 4.96·23-s − 2.68·24-s + 25-s + 1.09·26-s − 2.42·27-s + 3.03·28-s − 8.42·29-s + ⋯ |
| L(s) = 1 | + 0.219·2-s + 1.27·3-s − 0.951·4-s + 0.447·5-s + 0.281·6-s − 0.601·7-s − 0.429·8-s + 0.634·9-s + 0.0983·10-s + 0.395·11-s − 1.21·12-s + 0.977·13-s − 0.132·14-s + 0.571·15-s + 0.857·16-s + 0.139·18-s + 1.01·19-s − 0.425·20-s − 0.769·21-s + 0.0869·22-s + 1.03·23-s − 0.548·24-s + 0.200·25-s + 0.215·26-s − 0.467·27-s + 0.572·28-s − 1.56·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.540134109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.540134109\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.311T + 2T^{2} \) |
| 3 | \( 1 - 2.21T + 3T^{2} \) |
| 7 | \( 1 + 1.59T + 7T^{2} \) |
| 11 | \( 1 - 1.31T + 11T^{2} \) |
| 13 | \( 1 - 3.52T + 13T^{2} \) |
| 19 | \( 1 - 4.42T + 19T^{2} \) |
| 23 | \( 1 - 4.96T + 23T^{2} \) |
| 29 | \( 1 + 8.42T + 29T^{2} \) |
| 31 | \( 1 - 7.73T + 31T^{2} \) |
| 37 | \( 1 - 7.05T + 37T^{2} \) |
| 41 | \( 1 - 3.67T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 3.33T + 47T^{2} \) |
| 53 | \( 1 - 9.18T + 53T^{2} \) |
| 59 | \( 1 + 1.37T + 59T^{2} \) |
| 61 | \( 1 - 15.4T + 61T^{2} \) |
| 67 | \( 1 + 9.13T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 5.57T + 73T^{2} \) |
| 79 | \( 1 - 7.87T + 79T^{2} \) |
| 83 | \( 1 + 7.19T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
| 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
−9.472957450205911052178666138706, −8.816344605301828603035142672506, −8.188598267792485651625608484590, −7.24359808932287437442989331704, −6.14508775725909714245518178337, −5.36661343824730929765580468970, −4.15538079896070001771211144446, −3.45481291859106040056487329264, −2.68368124237410528312024851070, −1.13750599423387796901865828595,
1.13750599423387796901865828595, 2.68368124237410528312024851070, 3.45481291859106040056487329264, 4.15538079896070001771211144446, 5.36661343824730929765580468970, 6.14508775725909714245518178337, 7.24359808932287437442989331704, 8.188598267792485651625608484590, 8.816344605301828603035142672506, 9.472957450205911052178666138706
