| L(s) = 1 | − 1.64·3-s + 4.90·7-s − 0.284·9-s − 4.63·11-s + 1.40·13-s + 0.217·17-s + 1.04·19-s − 8.07·21-s − 7.50·23-s + 5.41·27-s + 3.93·29-s − 8.69·31-s + 7.64·33-s + 8.90·37-s − 2.30·39-s + 2.64·41-s + 6.67·43-s − 4.63·47-s + 17.0·49-s − 0.358·51-s − 1.36·53-s − 1.72·57-s − 13.2·59-s + 3.83·61-s − 1.39·63-s − 3.17·67-s + 12.3·69-s + ⋯ |
| L(s) = 1 | − 0.951·3-s + 1.85·7-s − 0.0948·9-s − 1.39·11-s + 0.388·13-s + 0.0527·17-s + 0.240·19-s − 1.76·21-s − 1.56·23-s + 1.04·27-s + 0.730·29-s − 1.56·31-s + 1.33·33-s + 1.46·37-s − 0.369·39-s + 0.413·41-s + 1.01·43-s − 0.676·47-s + 2.43·49-s − 0.0502·51-s − 0.187·53-s − 0.228·57-s − 1.72·59-s + 0.491·61-s − 0.175·63-s − 0.387·67-s + 1.48·69-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 | \( 1 + 1.64T + 3T^{2} \) |
| 7 | \( 1 - 4.90T + 7T^{2} \) |
| 11 | \( 1 + 4.63T + 11T^{2} \) |
| 13 | \( 1 - 1.40T + 13T^{2} \) |
| 17 | \( 1 - 0.217T + 17T^{2} \) |
| 19 | \( 1 - 1.04T + 19T^{2} \) |
| 23 | \( 1 + 7.50T + 23T^{2} \) |
| 29 | \( 1 - 3.93T + 29T^{2} \) |
| 31 | \( 1 + 8.69T + 31T^{2} \) |
| 37 | \( 1 - 8.90T + 37T^{2} \) |
| 41 | \( 1 - 2.64T + 41T^{2} \) |
| 43 | \( 1 - 6.67T + 43T^{2} \) |
| 47 | \( 1 + 4.63T + 47T^{2} \) |
| 53 | \( 1 + 1.36T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 3.83T + 61T^{2} \) |
| 67 | \( 1 + 3.17T + 67T^{2} \) |
| 71 | \( 1 - 4.22T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 4.73T + 79T^{2} \) |
| 83 | \( 1 - 7.72T + 83T^{2} \) |
| 89 | \( 1 + 9.58T + 89T^{2} \) |
| 97 | \( 1 + 5.34T + 97T^{2} \) |
| 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
−7.61786114192103312056149316564, −6.47421136630752844360421424196, −5.66068630664382402854241381822, −5.44251590187347544231410335632, −4.66251779228945976283158708818, −4.12781734929641419108312772718, −2.85713611974151000446898844094, −2.06048610986418441556865697553, −1.15167979928096524531538682065, 0,
1.15167979928096524531538682065, 2.06048610986418441556865697553, 2.85713611974151000446898844094, 4.12781734929641419108312772718, 4.66251779228945976283158708818, 5.44251590187347544231410335632, 5.66068630664382402854241381822, 6.47421136630752844360421424196, 7.61786114192103312056149316564
