| L(s) = 1 | − 2.73·3-s − 2·7-s + 4.46·9-s − 3.46·11-s + 2.73·13-s + 3.46·17-s + 19-s + 5.46·21-s + 3.46·23-s − 3.99·27-s + 3.46·29-s − 1.46·31-s + 9.46·33-s − 6.73·37-s − 7.46·39-s − 6·41-s + 4.92·43-s − 12.9·47-s − 3·49-s − 9.46·51-s + 10.7·53-s − 2.73·57-s + 6.92·59-s + 12.3·61-s − 8.92·63-s − 6.73·67-s − 9.46·69-s + ⋯ |
| L(s) = 1 | − 1.57·3-s − 0.755·7-s + 1.48·9-s − 1.04·11-s + 0.757·13-s + 0.840·17-s + 0.229·19-s + 1.19·21-s + 0.722·23-s − 0.769·27-s + 0.643·29-s − 0.262·31-s + 1.64·33-s − 1.10·37-s − 1.19·39-s − 0.937·41-s + 0.751·43-s − 1.88·47-s − 0.428·49-s − 1.32·51-s + 1.47·53-s − 0.361·57-s + 0.901·59-s + 1.58·61-s − 1.12·63-s − 0.822·67-s − 1.13·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 6.73T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 6.73T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 - 0.535T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
| 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
−8.855511533935697558017854457238, −7.938551033620986159704715607338, −6.92184041146793511736874997927, −6.44578648275118741177808487897, −5.39853823930531554967607062417, −5.21542994543241923041358277048, −3.89784477540040678313905319587, −2.89224244568729188193746483028, −1.23019904924336113712124714626, 0,
1.23019904924336113712124714626, 2.89224244568729188193746483028, 3.89784477540040678313905319587, 5.21542994543241923041358277048, 5.39853823930531554967607062417, 6.44578648275118741177808487897, 6.92184041146793511736874997927, 7.938551033620986159704715607338, 8.855511533935697558017854457238
