| L(s) = 1 | − 0.831·2-s + 3.34·3-s − 1.30·4-s + 5-s − 2.78·6-s − 2.41·7-s + 2.75·8-s + 8.19·9-s − 0.831·10-s + 2.18·11-s − 4.37·12-s + 3.36·13-s + 2.00·14-s + 3.34·15-s + 0.329·16-s − 2.40·17-s − 6.81·18-s − 1.03·19-s − 1.30·20-s − 8.08·21-s − 1.81·22-s + 1.61·23-s + 9.20·24-s + 25-s − 2.80·26-s + 17.3·27-s + 3.16·28-s + ⋯ |
| L(s) = 1 | − 0.587·2-s + 1.93·3-s − 0.654·4-s + 0.447·5-s − 1.13·6-s − 0.913·7-s + 0.972·8-s + 2.73·9-s − 0.262·10-s + 0.657·11-s − 1.26·12-s + 0.934·13-s + 0.537·14-s + 0.864·15-s + 0.0823·16-s − 0.583·17-s − 1.60·18-s − 0.237·19-s − 0.292·20-s − 1.76·21-s − 0.386·22-s + 0.337·23-s + 1.87·24-s + 0.200·25-s − 0.549·26-s + 3.34·27-s + 0.597·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.484025853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.484025853\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.831T + 2T^{2} \) |
| 3 | \( 1 - 3.34T + 3T^{2} \) |
| 7 | \( 1 + 2.41T + 7T^{2} \) |
| 11 | \( 1 - 2.18T + 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 + 2.40T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 - 1.61T + 23T^{2} \) |
| 29 | \( 1 - 6.63T + 29T^{2} \) |
| 31 | \( 1 + 5.05T + 31T^{2} \) |
| 37 | \( 1 + 1.24T + 37T^{2} \) |
| 41 | \( 1 + 4.42T + 41T^{2} \) |
| 43 | \( 1 - 5.87T + 43T^{2} \) |
| 47 | \( 1 - 1.42T + 47T^{2} \) |
| 53 | \( 1 + 4.97T + 53T^{2} \) |
| 59 | \( 1 + 8.97T + 59T^{2} \) |
| 61 | \( 1 - 9.78T + 61T^{2} \) |
| 67 | \( 1 - 8.39T + 67T^{2} \) |
| 71 | \( 1 - 4.78T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 - 3.18T + 79T^{2} \) |
| 83 | \( 1 - 3.45T + 83T^{2} \) |
| 89 | \( 1 + 3.60T + 89T^{2} \) |
| 97 | \( 1 - 2.44T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.017117479534724326501140634359, −8.678157226864372531783191059050, −7.942989065502222742383452666389, −7.00942112117082185111547191254, −6.33910534051293895303487712360, −4.82059113901518934961167531227, −3.89020193655773908426324123917, −3.33541463627773417032933019549, −2.19760881005609764085843569146, −1.16010441698693420739465173270,
1.16010441698693420739465173270, 2.19760881005609764085843569146, 3.33541463627773417032933019549, 3.89020193655773908426324123917, 4.82059113901518934961167531227, 6.33910534051293895303487712360, 7.00942112117082185111547191254, 7.942989065502222742383452666389, 8.678157226864372531783191059050, 9.017117479534724326501140634359
