| L(s) = 1 | − 3.08·3-s − 4.14·7-s + 6.52·9-s + 3.61·11-s − 2.37·13-s + 1.75·17-s − 1.28·19-s + 12.7·21-s − 2.23·23-s − 10.8·27-s − 1.43·29-s − 7.79·31-s − 11.1·33-s − 0.143·37-s + 7.33·39-s + 4.08·41-s + 11.7·43-s + 3.61·47-s + 10.1·49-s − 5.42·51-s − 9.61·53-s + 3.97·57-s − 6.47·59-s + 3.13·61-s − 27.0·63-s − 13.5·67-s + 6.88·69-s + ⋯ |
| L(s) = 1 | − 1.78·3-s − 1.56·7-s + 2.17·9-s + 1.08·11-s − 0.658·13-s + 0.426·17-s − 0.295·19-s + 2.79·21-s − 0.465·23-s − 2.09·27-s − 0.267·29-s − 1.40·31-s − 1.94·33-s − 0.0235·37-s + 1.17·39-s + 0.638·41-s + 1.79·43-s + 0.526·47-s + 1.45·49-s − 0.759·51-s − 1.32·53-s + 0.527·57-s − 0.842·59-s + 0.401·61-s − 3.40·63-s − 1.65·67-s + 0.829·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 3.08T + 3T^{2} \) |
| 7 | \( 1 + 4.14T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + 2.37T + 13T^{2} \) |
| 17 | \( 1 - 1.75T + 17T^{2} \) |
| 19 | \( 1 + 1.28T + 19T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 + 1.43T + 29T^{2} \) |
| 31 | \( 1 + 7.79T + 31T^{2} \) |
| 37 | \( 1 + 0.143T + 37T^{2} \) |
| 41 | \( 1 - 4.08T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 3.61T + 47T^{2} \) |
| 53 | \( 1 + 9.61T + 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 - 3.13T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + 5.78T + 73T^{2} \) |
| 79 | \( 1 + 8.19T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 5.64T + 89T^{2} \) |
| 97 | \( 1 + 5.88T + 97T^{2} \) |
| 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
−7.18278892146595896124931015856, −6.39844260700598685186631854800, −6.03395591652932175735370915674, −5.53886374121391941974600028415, −4.56457476358301098119512810009, −3.98357637475153732148680103317, −3.19234381295218340320529252971, −1.95169111218387897827589356288, −0.838985188576813983465885174598, 0,
0.838985188576813983465885174598, 1.95169111218387897827589356288, 3.19234381295218340320529252971, 3.98357637475153732148680103317, 4.56457476358301098119512810009, 5.53886374121391941974600028415, 6.03395591652932175735370915674, 6.39844260700598685186631854800, 7.18278892146595896124931015856
