| L(s) = 1 | + 2.39·2-s + 2.21·3-s + 3.74·4-s + 0.529·5-s + 5.30·6-s + 0.0274·7-s + 4.16·8-s + 1.90·9-s + 1.26·10-s − 11-s + 8.28·12-s − 0.760·13-s + 0.0657·14-s + 1.17·15-s + 2.50·16-s + 5.18·17-s + 4.56·18-s + 3.37·19-s + 1.98·20-s + 0.0607·21-s − 2.39·22-s − 7.91·23-s + 9.23·24-s − 4.71·25-s − 1.82·26-s − 2.42·27-s + 0.102·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.27·3-s + 1.87·4-s + 0.236·5-s + 2.16·6-s + 0.0103·7-s + 1.47·8-s + 0.635·9-s + 0.401·10-s − 0.301·11-s + 2.39·12-s − 0.211·13-s + 0.0175·14-s + 0.303·15-s + 0.627·16-s + 1.25·17-s + 1.07·18-s + 0.774·19-s + 0.443·20-s + 0.0132·21-s − 0.510·22-s − 1.65·23-s + 1.88·24-s − 0.943·25-s − 0.357·26-s − 0.466·27-s + 0.0193·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.704163740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.704163740\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 3 | \( 1 - 2.21T + 3T^{2} \) |
| 5 | \( 1 - 0.529T + 5T^{2} \) |
| 7 | \( 1 - 0.0274T + 7T^{2} \) |
| 13 | \( 1 + 0.760T + 13T^{2} \) |
| 17 | \( 1 - 5.18T + 17T^{2} \) |
| 19 | \( 1 - 3.37T + 19T^{2} \) |
| 23 | \( 1 + 7.91T + 23T^{2} \) |
| 29 | \( 1 + 3.81T + 29T^{2} \) |
| 31 | \( 1 - 0.922T + 31T^{2} \) |
| 37 | \( 1 - 3.05T + 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 + 3.83T + 43T^{2} \) |
| 47 | \( 1 - 8.14T + 47T^{2} \) |
| 53 | \( 1 - 9.06T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + 8.91T + 73T^{2} \) |
| 79 | \( 1 + 9.36T + 79T^{2} \) |
| 83 | \( 1 + 9.35T + 83T^{2} \) |
| 89 | \( 1 + 1.48T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
| 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
−9.730699184210179423423656602561, −8.534396518574039740036872703803, −7.76012671947466222957917850539, −7.10090229939952335303685562756, −5.81530315773066065574829671923, −5.45818748307006912979646704755, −4.15058889352757093562488311760, −3.56790609111328389994515199846, −2.70976821575125304508634902229, −1.89848725004371562156253756392,
1.89848725004371562156253756392, 2.70976821575125304508634902229, 3.56790609111328389994515199846, 4.15058889352757093562488311760, 5.45818748307006912979646704755, 5.81530315773066065574829671923, 7.10090229939952335303685562756, 7.76012671947466222957917850539, 8.534396518574039740036872703803, 9.730699184210179423423656602561
