| L(s) = 1 | − 2-s + 0.832·3-s + 4-s + 5-s − 0.832·6-s + 2.10·7-s − 8-s − 2.30·9-s − 10-s + 2.02·11-s + 0.832·12-s + 13-s − 2.10·14-s + 0.832·15-s + 16-s − 3.94·17-s + 2.30·18-s − 4.72·19-s + 20-s + 1.75·21-s − 2.02·22-s + 0.385·23-s − 0.832·24-s + 25-s − 26-s − 4.41·27-s + 2.10·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.480·3-s + 0.5·4-s + 0.447·5-s − 0.339·6-s + 0.794·7-s − 0.353·8-s − 0.769·9-s − 0.316·10-s + 0.612·11-s + 0.240·12-s + 0.277·13-s − 0.562·14-s + 0.214·15-s + 0.250·16-s − 0.956·17-s + 0.543·18-s − 1.08·19-s + 0.223·20-s + 0.382·21-s − 0.432·22-s + 0.0804·23-s − 0.169·24-s + 0.200·25-s − 0.196·26-s − 0.850·27-s + 0.397·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 - 0.832T + 3T^{2} \) |
| 7 | \( 1 - 2.10T + 7T^{2} \) |
| 11 | \( 1 - 2.02T + 11T^{2} \) |
| 17 | \( 1 + 3.94T + 17T^{2} \) |
| 19 | \( 1 + 4.72T + 19T^{2} \) |
| 23 | \( 1 - 0.385T + 23T^{2} \) |
| 29 | \( 1 + 9.67T + 29T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 3.10T + 41T^{2} \) |
| 43 | \( 1 + 0.987T + 43T^{2} \) |
| 47 | \( 1 + 5.76T + 47T^{2} \) |
| 53 | \( 1 - 7.93T + 53T^{2} \) |
| 59 | \( 1 + 7.05T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 4.50T + 67T^{2} \) |
| 71 | \( 1 + 6.87T + 71T^{2} \) |
| 73 | \( 1 + 3.90T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 6.16T + 83T^{2} \) |
| 89 | \( 1 - 4.75T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
| 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
−8.416589395711570567774691863859, −7.45239278101696961483535061394, −6.73713093654965303666953506401, −5.96353895313750413521607559948, −5.20363899415982268105680815526, −4.15101762328347899810531780413, −3.26445726605551549940619987038, −2.12535418103933782395065150549, −1.66952125940341429700425597174, 0,
1.66952125940341429700425597174, 2.12535418103933782395065150549, 3.26445726605551549940619987038, 4.15101762328347899810531780413, 5.20363899415982268105680815526, 5.96353895313750413521607559948, 6.73713093654965303666953506401, 7.45239278101696961483535061394, 8.416589395711570567774691863859
