| L(s) = 1 | − 5-s + 4·7-s − 11-s − 2·13-s − 2·19-s − 9·23-s − 4·25-s − 4·29-s + 5·31-s − 4·35-s − 9·37-s − 2·41-s − 6·43-s + 4·47-s + 9·49-s + 6·53-s + 55-s + 5·59-s + 2·65-s − 13·67-s + 71-s + 14·73-s − 4·77-s − 10·79-s − 14·83-s + 13·89-s − 8·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.301·11-s − 0.554·13-s − 0.458·19-s − 1.87·23-s − 4/5·25-s − 0.742·29-s + 0.898·31-s − 0.676·35-s − 1.47·37-s − 0.312·41-s − 0.914·43-s + 0.583·47-s + 9/7·49-s + 0.824·53-s + 0.134·55-s + 0.650·59-s + 0.248·65-s − 1.58·67-s + 0.118·71-s + 1.63·73-s − 0.455·77-s − 1.12·79-s − 1.53·83-s + 1.37·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
| 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
−8.186354510829608383904704605143, −7.74108127292070035304797363645, −6.98120863432558700165033152653, −5.89389826191186091054742635191, −5.17787900439164353057446639822, −4.40102075584760445465304755194, −3.73023548092803462605626492102, −2.35899421853380604273872472408, −1.63562494428470881571804048787, 0,
1.63562494428470881571804048787, 2.35899421853380604273872472408, 3.73023548092803462605626492102, 4.40102075584760445465304755194, 5.17787900439164353057446639822, 5.89389826191186091054742635191, 6.98120863432558700165033152653, 7.74108127292070035304797363645, 8.186354510829608383904704605143
