| L(s) = 1 | − 0.630·2-s + 3-s − 1.60·4-s − 2.79·5-s − 0.630·6-s − 7-s + 2.27·8-s + 9-s + 1.76·10-s + 1.05·11-s − 1.60·12-s − 2.76·13-s + 0.630·14-s − 2.79·15-s + 1.77·16-s − 0.136·17-s − 0.630·18-s − 0.161·19-s + 4.48·20-s − 21-s − 0.663·22-s − 0.925·23-s + 2.27·24-s + 2.83·25-s + 1.73·26-s + 27-s + 1.60·28-s + ⋯ |
| L(s) = 1 | − 0.445·2-s + 0.577·3-s − 0.801·4-s − 1.25·5-s − 0.257·6-s − 0.377·7-s + 0.802·8-s + 0.333·9-s + 0.557·10-s + 0.317·11-s − 0.462·12-s − 0.765·13-s + 0.168·14-s − 0.722·15-s + 0.443·16-s − 0.0331·17-s − 0.148·18-s − 0.0371·19-s + 1.00·20-s − 0.218·21-s − 0.141·22-s − 0.192·23-s + 0.463·24-s + 0.566·25-s + 0.341·26-s + 0.192·27-s + 0.302·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6908193066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6908193066\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 + 0.630T + 2T^{2} \) |
| 5 | \( 1 + 2.79T + 5T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 13 | \( 1 + 2.76T + 13T^{2} \) |
| 17 | \( 1 + 0.136T + 17T^{2} \) |
| 19 | \( 1 + 0.161T + 19T^{2} \) |
| 23 | \( 1 + 0.925T + 23T^{2} \) |
| 29 | \( 1 + 0.490T + 29T^{2} \) |
| 31 | \( 1 - 5.37T + 31T^{2} \) |
| 37 | \( 1 + 7.86T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 8.97T + 43T^{2} \) |
| 47 | \( 1 - 3.64T + 47T^{2} \) |
| 53 | \( 1 + 9.32T + 53T^{2} \) |
| 59 | \( 1 + 2.93T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 5.60T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 1.58T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.366837023137477377508318888064, −7.965730416412495643519933405626, −7.20838313825383748690580788936, −6.54496111178586373179324792877, −5.17267976192132891909461465656, −4.58566913995856019055738685975, −3.73418588077437566855250681813, −3.23481118086082725149085471022, −1.86035408187736603854218987210, −0.49219532977802163417225554905,
0.49219532977802163417225554905, 1.86035408187736603854218987210, 3.23481118086082725149085471022, 3.73418588077437566855250681813, 4.58566913995856019055738685975, 5.17267976192132891909461465656, 6.54496111178586373179324792877, 7.20838313825383748690580788936, 7.965730416412495643519933405626, 8.366837023137477377508318888064
