| L(s) = 1 | + 0.547·2-s − 3-s − 1.70·4-s − 0.955·5-s − 0.547·6-s − 4.37·7-s − 2.02·8-s + 9-s − 0.522·10-s + 11-s + 1.70·12-s − 2.39·14-s + 0.955·15-s + 2.29·16-s − 6.30·17-s + 0.547·18-s + 7.97·19-s + 1.62·20-s + 4.37·21-s + 0.547·22-s + 6.15·23-s + 2.02·24-s − 4.08·25-s − 27-s + 7.43·28-s + 6.17·29-s + 0.522·30-s + ⋯ |
| L(s) = 1 | + 0.386·2-s − 0.577·3-s − 0.850·4-s − 0.427·5-s − 0.223·6-s − 1.65·7-s − 0.716·8-s + 0.333·9-s − 0.165·10-s + 0.301·11-s + 0.490·12-s − 0.639·14-s + 0.246·15-s + 0.573·16-s − 1.52·17-s + 0.128·18-s + 1.82·19-s + 0.363·20-s + 0.954·21-s + 0.116·22-s + 1.28·23-s + 0.413·24-s − 0.817·25-s − 0.192·27-s + 1.40·28-s + 1.14·29-s + 0.0954·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.547T + 2T^{2} \) |
| 5 | \( 1 + 0.955T + 5T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 17 | \( 1 + 6.30T + 17T^{2} \) |
| 19 | \( 1 - 7.97T + 19T^{2} \) |
| 23 | \( 1 - 6.15T + 23T^{2} \) |
| 29 | \( 1 - 6.17T + 29T^{2} \) |
| 31 | \( 1 + 2.39T + 31T^{2} \) |
| 37 | \( 1 + 4.57T + 37T^{2} \) |
| 41 | \( 1 + 3.13T + 41T^{2} \) |
| 43 | \( 1 - 5.21T + 43T^{2} \) |
| 47 | \( 1 + 3.67T + 47T^{2} \) |
| 53 | \( 1 - 0.617T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 0.890T + 61T^{2} \) |
| 67 | \( 1 + 3.32T + 67T^{2} \) |
| 71 | \( 1 - 1.81T + 71T^{2} \) |
| 73 | \( 1 - 8.42T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 7.46T + 97T^{2} \) |
| 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
−7.65873318185965849644298510861, −6.78702405512356406786191021090, −6.46329851620926802796782353591, −5.47935116556040661229403086906, −4.96872017335169985857214258251, −3.99406343161546585088691406607, −3.49813673410896424660288369599, −2.67851823150207186422766398231, −0.933377360368098984859495133152, 0,
0.933377360368098984859495133152, 2.67851823150207186422766398231, 3.49813673410896424660288369599, 3.99406343161546585088691406607, 4.96872017335169985857214258251, 5.47935116556040661229403086906, 6.46329851620926802796782353591, 6.78702405512356406786191021090, 7.65873318185965849644298510861
