| L(s) = 1 | + 147.·7-s − 662.·11-s + 629.·13-s + 1.08e3·17-s − 2.52e3·19-s + 1.80e3·23-s − 6.23e3·29-s + 5.61e3·31-s + 1.19e3·37-s − 1.00e4·41-s − 1.07e4·43-s + 3.80e3·47-s + 5.02e3·49-s − 1.58e4·53-s − 1.73e3·59-s + 1.11e4·61-s + 4.86e4·67-s − 6.89e4·71-s − 8.06e4·73-s − 9.78e4·77-s + 8.20e4·79-s + 8.56e4·83-s − 2.58e4·89-s + 9.30e4·91-s − 9.93e3·97-s − 4.41e4·101-s + 1.31e5·103-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 1.65·11-s + 1.03·13-s + 0.908·17-s − 1.60·19-s + 0.710·23-s − 1.37·29-s + 1.04·31-s + 0.144·37-s − 0.930·41-s − 0.885·43-s + 0.251·47-s + 0.298·49-s − 0.772·53-s − 0.0648·59-s + 0.385·61-s + 1.32·67-s − 1.62·71-s − 1.77·73-s − 1.88·77-s + 1.47·79-s + 1.36·83-s − 0.345·89-s + 1.17·91-s − 0.107·97-s − 0.430·101-s + 1.22·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 147.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 662.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 629.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.08e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.52e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.23e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.19e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.80e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.58e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.73e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.06e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.56e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.58e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.93e3T + 8.58e9T^{2} \) |
| 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
−8.651019904189100464906892614887, −8.183869995799523271064795447888, −7.46283491804060302503512644355, −6.26696545416944167129545242941, −5.34240824882673633972297664976, −4.65346651795094176677449636682, −3.47341994619888123025227796589, −2.31667957692507180839250479226, −1.32949060373189231350106413096, 0,
1.32949060373189231350106413096, 2.31667957692507180839250479226, 3.47341994619888123025227796589, 4.65346651795094176677449636682, 5.34240824882673633972297664976, 6.26696545416944167129545242941, 7.46283491804060302503512644355, 8.183869995799523271064795447888, 8.651019904189100464906892614887
