| L(s) = 1 | + 8.54·2-s + 29.2·3-s + 41.0·4-s + 25·5-s + 249.·6-s + 146.·7-s + 77.0·8-s + 610.·9-s + 213.·10-s − 121·11-s + 1.19e3·12-s + 643.·13-s + 1.25e3·14-s + 730.·15-s − 654.·16-s − 518.·17-s + 5.21e3·18-s − 361·19-s + 1.02e3·20-s + 4.27e3·21-s − 1.03e3·22-s − 3.63e3·23-s + 2.25e3·24-s + 625·25-s + 5.49e3·26-s + 1.07e4·27-s + 6.00e3·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.87·3-s + 1.28·4-s + 0.447·5-s + 2.83·6-s + 1.12·7-s + 0.425·8-s + 2.51·9-s + 0.675·10-s − 0.301·11-s + 2.40·12-s + 1.05·13-s + 1.70·14-s + 0.837·15-s − 0.638·16-s − 0.435·17-s + 3.79·18-s − 0.229·19-s + 0.573·20-s + 2.11·21-s − 0.455·22-s − 1.43·23-s + 0.797·24-s + 0.200·25-s + 1.59·26-s + 2.83·27-s + 1.44·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(14.89474111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.89474111\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 8.54T + 32T^{2} \) |
| 3 | \( 1 - 29.2T + 243T^{2} \) |
| 7 | \( 1 - 146.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 643.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 518.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.63e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 765.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.65e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.82e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.08e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.28e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.66e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 9.15e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.82e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.41e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.18e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.63e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.11e4T + 8.58e9T^{2} \) |
| 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
−8.824214379726697011976841915458, −8.467464311128879961645312563620, −7.55704876318377190638913095301, −6.57678596061215521176201005248, −5.57643817987602868907815289145, −4.43443158145169249913314617740, −4.03143483797251067571083018209, −2.95647212163873317655354549247, −2.24129178270147351942688743066, −1.43923650272519771775722048403,
1.43923650272519771775722048403, 2.24129178270147351942688743066, 2.95647212163873317655354549247, 4.03143483797251067571083018209, 4.43443158145169249913314617740, 5.57643817987602868907815289145, 6.57678596061215521176201005248, 7.55704876318377190638913095301, 8.467464311128879961645312563620, 8.824214379726697011976841915458
