| L(s) = 1 | − 0.167·2-s + 3-s − 1.97·4-s + 1.16·5-s − 0.167·6-s − 7-s + 0.665·8-s + 9-s − 0.195·10-s − 1.80·11-s − 1.97·12-s + 6.97·13-s + 0.167·14-s + 1.16·15-s + 3.83·16-s + 6.13·17-s − 0.167·18-s + 1.80·19-s − 2.30·20-s − 21-s + 0.302·22-s − 23-s + 0.665·24-s − 3.63·25-s − 1.16·26-s + 27-s + 1.97·28-s + ⋯ |
| L(s) = 1 | − 0.118·2-s + 0.577·3-s − 0.985·4-s + 0.522·5-s − 0.0683·6-s − 0.377·7-s + 0.235·8-s + 0.333·9-s − 0.0618·10-s − 0.544·11-s − 0.569·12-s + 1.93·13-s + 0.0447·14-s + 0.301·15-s + 0.958·16-s + 1.48·17-s − 0.0394·18-s + 0.413·19-s − 0.514·20-s − 0.218·21-s + 0.0644·22-s − 0.208·23-s + 0.135·24-s − 0.727·25-s − 0.228·26-s + 0.192·27-s + 0.372·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.485448214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.485448214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.167T + 2T^{2} \) |
| 5 | \( 1 - 1.16T + 5T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 - 6.97T + 13T^{2} \) |
| 17 | \( 1 - 6.13T + 17T^{2} \) |
| 19 | \( 1 - 1.80T + 19T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 + 2.19T + 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 - 9.74T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 + 1.94T + 47T^{2} \) |
| 53 | \( 1 - 1.16T + 53T^{2} \) |
| 59 | \( 1 - 9.11T + 59T^{2} \) |
| 61 | \( 1 + 4.63T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 + 9.25T + 71T^{2} \) |
| 73 | \( 1 + 0.474T + 73T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 - 8.13T + 83T^{2} \) |
| 89 | \( 1 - 1.02T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
| 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
−10.62770955364142862647830192578, −9.969464385981987051860810704284, −9.197089042058381749106935282653, −8.378429271904773033536201663148, −7.66848761868815121323227590754, −6.16203618332346767751848145449, −5.38964569241539682853565737802, −4.00068566652171835500489046491, −3.13484636625349900617561674732, −1.27326967760055291313327154889,
1.27326967760055291313327154889, 3.13484636625349900617561674732, 4.00068566652171835500489046491, 5.38964569241539682853565737802, 6.16203618332346767751848145449, 7.66848761868815121323227590754, 8.378429271904773033536201663148, 9.197089042058381749106935282653, 9.969464385981987051860810704284, 10.62770955364142862647830192578
