| L(s) = 1 | + 2.80·2-s − 0.198·3-s + 5.85·4-s + 2.49·5-s − 0.554·6-s + 10.7·8-s − 2.96·9-s + 6.98·10-s − 1.10·11-s − 1.15·12-s − 5.15·13-s − 0.493·15-s + 18.5·16-s + 6.45·17-s − 8.29·18-s + 1.86·19-s + 14.5·20-s − 3.10·22-s − 3.08·23-s − 2.13·24-s + 1.21·25-s − 14.4·26-s + 1.18·27-s + 0.219·29-s − 1.38·30-s + 0.670·31-s + 30.3·32-s + ⋯ |
| L(s) = 1 | + 1.98·2-s − 0.114·3-s + 2.92·4-s + 1.11·5-s − 0.226·6-s + 3.81·8-s − 0.986·9-s + 2.20·10-s − 0.334·11-s − 0.334·12-s − 1.43·13-s − 0.127·15-s + 4.63·16-s + 1.56·17-s − 1.95·18-s + 0.427·19-s + 3.26·20-s − 0.663·22-s − 0.643·23-s − 0.436·24-s + 0.243·25-s − 2.83·26-s + 0.227·27-s + 0.0408·29-s − 0.252·30-s + 0.120·31-s + 5.36·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.993599451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.993599451\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 2.80T + 2T^{2} \) |
| 3 | \( 1 + 0.198T + 3T^{2} \) |
| 5 | \( 1 - 2.49T + 5T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 + 5.15T + 13T^{2} \) |
| 17 | \( 1 - 6.45T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 + 3.08T + 23T^{2} \) |
| 29 | \( 1 - 0.219T + 29T^{2} \) |
| 31 | \( 1 - 0.670T + 31T^{2} \) |
| 37 | \( 1 - 1.33T + 37T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 4.44T + 47T^{2} \) |
| 53 | \( 1 + 6.89T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 6.27T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 6.98T + 71T^{2} \) |
| 73 | \( 1 + 5.20T + 73T^{2} \) |
| 79 | \( 1 - 0.792T + 79T^{2} \) |
| 83 | \( 1 - 8.76T + 83T^{2} \) |
| 89 | \( 1 + 9.70T + 89T^{2} \) |
| 97 | \( 1 + 8.29T + 97T^{2} \) |
| 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
−9.446185006130498578274081979841, −7.85468553290404375724846619768, −7.42635240143945743714368258746, −6.22797098980346550870837753264, −5.80237862850420485691593078890, −5.23178742726866875793296207504, −4.48163669275998840225861502969, −3.15680495958965751677223592807, −2.67889034809801640955463109591, −1.66334054066544373135687737444,
1.66334054066544373135687737444, 2.67889034809801640955463109591, 3.15680495958965751677223592807, 4.48163669275998840225861502969, 5.23178742726866875793296207504, 5.80237862850420485691593078890, 6.22797098980346550870837753264, 7.42635240143945743714368258746, 7.85468553290404375724846619768, 9.446185006130498578274081979841
