| L(s) = 1 | + 1.35·2-s − 3-s − 0.164·4-s + 2.92·5-s − 1.35·6-s − 0.124·7-s − 2.93·8-s + 9-s + 3.95·10-s − 6.11·11-s + 0.164·12-s + 4.37·13-s − 0.169·14-s − 2.92·15-s − 3.64·16-s − 17-s + 1.35·18-s + 4.28·19-s − 0.480·20-s + 0.124·21-s − 8.28·22-s + 5.01·23-s + 2.93·24-s + 3.54·25-s + 5.93·26-s − 27-s + 0.0205·28-s + ⋯ |
| L(s) = 1 | + 0.958·2-s − 0.577·3-s − 0.0821·4-s + 1.30·5-s − 0.553·6-s − 0.0472·7-s − 1.03·8-s + 0.333·9-s + 1.25·10-s − 1.84·11-s + 0.0474·12-s + 1.21·13-s − 0.0452·14-s − 0.754·15-s − 0.911·16-s − 0.242·17-s + 0.319·18-s + 0.982·19-s − 0.107·20-s + 0.0272·21-s − 1.76·22-s + 1.04·23-s + 0.598·24-s + 0.708·25-s + 1.16·26-s − 0.192·27-s + 0.00387·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.657263265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.657263265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 1.35T + 2T^{2} \) |
| 5 | \( 1 - 2.92T + 5T^{2} \) |
| 7 | \( 1 + 0.124T + 7T^{2} \) |
| 11 | \( 1 + 6.11T + 11T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 19 | \( 1 - 4.28T + 19T^{2} \) |
| 23 | \( 1 - 5.01T + 23T^{2} \) |
| 29 | \( 1 + 4.63T + 29T^{2} \) |
| 31 | \( 1 + 6.51T + 31T^{2} \) |
| 37 | \( 1 - 4.62T + 37T^{2} \) |
| 41 | \( 1 - 1.36T + 41T^{2} \) |
| 43 | \( 1 - 7.36T + 43T^{2} \) |
| 47 | \( 1 + 0.295T + 47T^{2} \) |
| 53 | \( 1 - 7.60T + 53T^{2} \) |
| 59 | \( 1 + 5.33T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 2.08T + 67T^{2} \) |
| 71 | \( 1 + 3.92T + 71T^{2} \) |
| 73 | \( 1 + 3.36T + 73T^{2} \) |
| 79 | \( 1 - 5.61T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 2.21T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−7.66424789989581431108978832368, −6.92605498837465441208847695738, −5.95559665613712259951520376715, −5.68671757627320959461626731796, −5.24709246385969040367675917855, −4.52979102215628608917780398944, −3.52568853950274873853525307093, −2.81581216166435941528146997170, −1.95172455150328739015875587741, −0.71109565185780038852758815568,
0.71109565185780038852758815568, 1.95172455150328739015875587741, 2.81581216166435941528146997170, 3.52568853950274873853525307093, 4.52979102215628608917780398944, 5.24709246385969040367675917855, 5.68671757627320959461626731796, 5.95559665613712259951520376715, 6.92605498837465441208847695738, 7.66424789989581431108978832368
