| L(s) = 1 | − 2-s + 1.51·3-s + 4-s + 2.10·5-s − 1.51·6-s + 2.35·7-s − 8-s − 0.695·9-s − 2.10·10-s − 1.36·11-s + 1.51·12-s + 6.92·13-s − 2.35·14-s + 3.19·15-s + 16-s − 0.171·17-s + 0.695·18-s − 6.80·19-s + 2.10·20-s + 3.57·21-s + 1.36·22-s − 1.85·23-s − 1.51·24-s − 0.561·25-s − 6.92·26-s − 5.61·27-s + 2.35·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.876·3-s + 0.5·4-s + 0.942·5-s − 0.619·6-s + 0.890·7-s − 0.353·8-s − 0.231·9-s − 0.666·10-s − 0.410·11-s + 0.438·12-s + 1.92·13-s − 0.629·14-s + 0.825·15-s + 0.250·16-s − 0.0415·17-s + 0.164·18-s − 1.56·19-s + 0.471·20-s + 0.780·21-s + 0.290·22-s − 0.387·23-s − 0.309·24-s − 0.112·25-s − 1.35·26-s − 1.07·27-s + 0.445·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.653742470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.653742470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 223 | \( 1 - T \) |
| good | 3 | \( 1 - 1.51T + 3T^{2} \) |
| 5 | \( 1 - 2.10T + 5T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 11 | \( 1 + 1.36T + 11T^{2} \) |
| 13 | \( 1 - 6.92T + 13T^{2} \) |
| 17 | \( 1 + 0.171T + 17T^{2} \) |
| 19 | \( 1 + 6.80T + 19T^{2} \) |
| 23 | \( 1 + 1.85T + 23T^{2} \) |
| 29 | \( 1 + 1.03T + 29T^{2} \) |
| 31 | \( 1 - 7.57T + 31T^{2} \) |
| 37 | \( 1 - 0.829T + 37T^{2} \) |
| 41 | \( 1 + 4.45T + 41T^{2} \) |
| 43 | \( 1 - 2.68T + 43T^{2} \) |
| 47 | \( 1 - 9.80T + 47T^{2} \) |
| 53 | \( 1 - 4.58T + 53T^{2} \) |
| 59 | \( 1 + 8.54T + 59T^{2} \) |
| 61 | \( 1 - 4.85T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 1.03T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 7.16T + 83T^{2} \) |
| 89 | \( 1 - 5.27T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−10.85923634128630468566352130053, −10.18983412564441544631808168309, −9.019228057526919081565653726625, −8.498805031272161992276678049134, −7.896103514483677959074297554455, −6.41828218568318770191882112914, −5.67362849428022658237929635838, −4.05440854339232306654298546121, −2.59492525606021058242017575541, −1.60900304849293628742217475971,
1.60900304849293628742217475971, 2.59492525606021058242017575541, 4.05440854339232306654298546121, 5.67362849428022658237929635838, 6.41828218568318770191882112914, 7.896103514483677959074297554455, 8.498805031272161992276678049134, 9.019228057526919081565653726625, 10.18983412564441544631808168309, 10.85923634128630468566352130053
