L(s) = 1 | − 2.73·3-s − 5-s − 3.46·7-s + 4.46·9-s + 3.46·11-s − 6.73·13-s + 2.73·15-s + 2.73·17-s + 6.92·19-s + 9.46·21-s + 8.19·23-s + 25-s − 3.99·27-s + 5.46·29-s − 5.46·31-s − 9.46·33-s + 3.46·35-s − 0.196·37-s + 18.3·39-s − 12.3·41-s − 2·43-s − 4.46·45-s − 8.92·47-s + 4.99·49-s − 7.46·51-s + 12.1·53-s − 3.46·55-s + ⋯ |
L(s) = 1 | − 1.57·3-s − 0.447·5-s − 1.30·7-s + 1.48·9-s + 1.04·11-s − 1.86·13-s + 0.705·15-s + 0.662·17-s + 1.58·19-s + 2.06·21-s + 1.70·23-s + 0.200·25-s − 0.769·27-s + 1.01·29-s − 0.981·31-s − 1.64·33-s + 0.585·35-s − 0.0322·37-s + 2.94·39-s − 1.93·41-s − 0.304·43-s − 0.665·45-s − 1.30·47-s + 0.714·49-s − 1.04·51-s + 1.67·53-s − 0.467·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5881669100\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5881669100\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 6.73T + 13T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 + 0.196T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 8.92T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 8.19T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 0.928T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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
−7.29684120061152659578309552910, −7.06325261747275529071148803104, −6.59851075117276055085247849550, −5.64971065936200033759038193575, −5.14133464191926448945115820768, −4.55854777703623690564683632452, −3.44482908635180680346859506415, −2.94462329159905247246810262698, −1.35341986370336354540169973606, −0.44951363442382159503491089694,
0.44951363442382159503491089694, 1.35341986370336354540169973606, 2.94462329159905247246810262698, 3.44482908635180680346859506415, 4.55854777703623690564683632452, 5.14133464191926448945115820768, 5.64971065936200033759038193575, 6.59851075117276055085247849550, 7.06325261747275529071148803104, 7.29684120061152659578309552910