| L(s) = 1 | + 1.30·3-s + 0.966·5-s − 1.29·9-s + 3.78·11-s − 2.23·13-s + 1.26·15-s + 2.45·17-s − 8.23·19-s − 4.87·23-s − 4.06·25-s − 5.60·27-s + 2.43·29-s − 1.65·31-s + 4.94·33-s − 3.64·37-s − 2.91·39-s − 0.608·41-s − 3.24·43-s − 1.24·45-s − 9.46·47-s + 3.20·51-s − 1.76·53-s + 3.66·55-s − 10.7·57-s + 0.605·59-s + 8.41·61-s − 2.15·65-s + ⋯ |
| L(s) = 1 | + 0.754·3-s + 0.432·5-s − 0.430·9-s + 1.14·11-s − 0.619·13-s + 0.326·15-s + 0.594·17-s − 1.89·19-s − 1.01·23-s − 0.813·25-s − 1.07·27-s + 0.452·29-s − 0.296·31-s + 0.861·33-s − 0.599·37-s − 0.467·39-s − 0.0950·41-s − 0.495·43-s − 0.186·45-s − 1.38·47-s + 0.448·51-s − 0.242·53-s + 0.493·55-s − 1.42·57-s + 0.0787·59-s + 1.07·61-s − 0.267·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 5 | \( 1 - 0.966T + 5T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 13 | \( 1 + 2.23T + 13T^{2} \) |
| 17 | \( 1 - 2.45T + 17T^{2} \) |
| 19 | \( 1 + 8.23T + 19T^{2} \) |
| 23 | \( 1 + 4.87T + 23T^{2} \) |
| 29 | \( 1 - 2.43T + 29T^{2} \) |
| 31 | \( 1 + 1.65T + 31T^{2} \) |
| 37 | \( 1 + 3.64T + 37T^{2} \) |
| 41 | \( 1 + 0.608T + 41T^{2} \) |
| 43 | \( 1 + 3.24T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 + 1.76T + 53T^{2} \) |
| 59 | \( 1 - 0.605T + 59T^{2} \) |
| 61 | \( 1 - 8.41T + 61T^{2} \) |
| 67 | \( 1 + 9.34T + 67T^{2} \) |
| 71 | \( 1 - 5.91T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 9.28T + 97T^{2} \) |
| 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
−7.958290478991987881811775109270, −7.12889488925622187816493116628, −6.27742026967340984265161641145, −5.85461581149625415850680327808, −4.76866120702434108663437985407, −3.97335696079061665377357288266, −3.27855274137526227207923349193, −2.24939349358846530095648096134, −1.68883297549358668712299571920, 0,
1.68883297549358668712299571920, 2.24939349358846530095648096134, 3.27855274137526227207923349193, 3.97335696079061665377357288266, 4.76866120702434108663437985407, 5.85461581149625415850680327808, 6.27742026967340984265161641145, 7.12889488925622187816493116628, 7.958290478991987881811775109270
