| L(s) = 1 | + 0.219·2-s + 1.60·3-s − 1.95·4-s + 0.351·6-s + 0.332·7-s − 0.868·8-s − 0.439·9-s − 5.37·11-s − 3.12·12-s + 0.0729·14-s + 3.71·16-s + 5.06·17-s − 0.0965·18-s + 2.26·19-s + 0.531·21-s − 1.18·22-s + 2.83·23-s − 1.38·24-s − 5.50·27-s − 0.648·28-s − 2.90·29-s + 5.46·31-s + 2.55·32-s − 8.59·33-s + 1.11·34-s + 0.857·36-s − 5.97·37-s + ⋯ |
| L(s) = 1 | + 0.155·2-s + 0.923·3-s − 0.975·4-s + 0.143·6-s + 0.125·7-s − 0.306·8-s − 0.146·9-s − 1.61·11-s − 0.901·12-s + 0.0195·14-s + 0.928·16-s + 1.22·17-s − 0.0227·18-s + 0.520·19-s + 0.116·21-s − 0.251·22-s + 0.592·23-s − 0.283·24-s − 1.05·27-s − 0.122·28-s − 0.539·29-s + 0.981·31-s + 0.451·32-s − 1.49·33-s + 0.190·34-s + 0.142·36-s − 0.981·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.834558568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.834558568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.219T + 2T^{2} \) |
| 3 | \( 1 - 1.60T + 3T^{2} \) |
| 7 | \( 1 - 0.332T + 7T^{2} \) |
| 11 | \( 1 + 5.37T + 11T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 - 2.26T + 19T^{2} \) |
| 23 | \( 1 - 2.83T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 5.97T + 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 - 5.06T + 43T^{2} \) |
| 47 | \( 1 + 8.34T + 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 - 2.70T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 9.68T + 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 - 4.26T + 83T^{2} \) |
| 89 | \( 1 - 3.22T + 89T^{2} \) |
| 97 | \( 1 - 2.50T + 97T^{2} \) |
| 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
−8.215598351319461238884186749965, −8.002313615481687514612614036374, −7.25500620083369787390675465520, −5.93954391346405088334786803038, −5.28009847982693619413447572296, −4.76909852104274917697264579268, −3.53428883437294555433550297231, −3.15550906216449768702011007655, −2.18541160804132786755557800103, −0.70791127681211863382592782506,
0.70791127681211863382592782506, 2.18541160804132786755557800103, 3.15550906216449768702011007655, 3.53428883437294555433550297231, 4.76909852104274917697264579268, 5.28009847982693619413447572296, 5.93954391346405088334786803038, 7.25500620083369787390675465520, 8.002313615481687514612614036374, 8.215598351319461238884186749965
