| L(s) = 1 | − 0.436·2-s − 0.0103·3-s − 1.80·4-s + 3.47·5-s + 0.00452·6-s − 1.93·7-s + 1.66·8-s − 2.99·9-s − 1.51·10-s + 11-s + 0.0187·12-s + 4.56·13-s + 0.845·14-s − 0.0360·15-s + 2.89·16-s + 2.96·17-s + 1.31·18-s − 4.35·19-s − 6.29·20-s + 0.0200·21-s − 0.436·22-s + 6.48·23-s − 0.0172·24-s + 7.10·25-s − 1.99·26-s + 0.0622·27-s + 3.50·28-s + ⋯ |
| L(s) = 1 | − 0.308·2-s − 0.00598·3-s − 0.904·4-s + 1.55·5-s + 0.00184·6-s − 0.731·7-s + 0.588·8-s − 0.999·9-s − 0.480·10-s + 0.301·11-s + 0.00541·12-s + 1.26·13-s + 0.225·14-s − 0.00931·15-s + 0.722·16-s + 0.719·17-s + 0.308·18-s − 0.999·19-s − 1.40·20-s + 0.00437·21-s − 0.0931·22-s + 1.35·23-s − 0.00352·24-s + 1.42·25-s − 0.390·26-s + 0.0119·27-s + 0.661·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.280362551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280362551\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.436T + 2T^{2} \) |
| 3 | \( 1 + 0.0103T + 3T^{2} \) |
| 5 | \( 1 - 3.47T + 5T^{2} \) |
| 7 | \( 1 + 1.93T + 7T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 - 2.96T + 17T^{2} \) |
| 19 | \( 1 + 4.35T + 19T^{2} \) |
| 23 | \( 1 - 6.48T + 23T^{2} \) |
| 29 | \( 1 - 8.12T + 29T^{2} \) |
| 31 | \( 1 + 7.92T + 31T^{2} \) |
| 37 | \( 1 + 1.05T + 37T^{2} \) |
| 41 | \( 1 - 9.87T + 41T^{2} \) |
| 43 | \( 1 - 0.495T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 1.04T + 59T^{2} \) |
| 67 | \( 1 - 6.10T + 67T^{2} \) |
| 71 | \( 1 - 0.0603T + 71T^{2} \) |
| 73 | \( 1 - 2.52T + 73T^{2} \) |
| 79 | \( 1 - 5.40T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 18.5T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−10.43629368972825634962029569297, −9.390583198325121801775907196256, −9.033392733312696652738283666981, −8.281015338352045001754989530201, −6.76555432975489329446767906374, −5.90251859377501246591774675937, −5.33559812688345468593447029023, −3.88798763608955491958537779365, −2.69030356310863829064462665035, −1.08874159884096397522679018029,
1.08874159884096397522679018029, 2.69030356310863829064462665035, 3.88798763608955491958537779365, 5.33559812688345468593447029023, 5.90251859377501246591774675937, 6.76555432975489329446767906374, 8.281015338352045001754989530201, 9.033392733312696652738283666981, 9.390583198325121801775907196256, 10.43629368972825634962029569297
