| L(s) = 1 | + 1.24·2-s − 0.444·4-s + 1.99·5-s − 4.26·7-s − 3.04·8-s + 2.48·10-s + 2.17·11-s + 0.563·13-s − 5.31·14-s − 2.91·16-s + 2.08·17-s + 7.02·19-s − 0.885·20-s + 2.71·22-s − 23-s − 1.02·25-s + 0.702·26-s + 1.89·28-s − 29-s − 5.22·31-s + 2.46·32-s + 2.59·34-s − 8.49·35-s − 3.21·37-s + 8.76·38-s − 6.07·40-s − 4.83·41-s + ⋯ |
| L(s) = 1 | + 0.882·2-s − 0.222·4-s + 0.891·5-s − 1.61·7-s − 1.07·8-s + 0.786·10-s + 0.657·11-s + 0.156·13-s − 1.42·14-s − 0.728·16-s + 0.504·17-s + 1.61·19-s − 0.197·20-s + 0.579·22-s − 0.208·23-s − 0.205·25-s + 0.137·26-s + 0.357·28-s − 0.185·29-s − 0.939·31-s + 0.435·32-s + 0.445·34-s − 1.43·35-s − 0.528·37-s + 1.42·38-s − 0.960·40-s − 0.755·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.24T + 2T^{2} \) |
| 5 | \( 1 - 1.99T + 5T^{2} \) |
| 7 | \( 1 + 4.26T + 7T^{2} \) |
| 11 | \( 1 - 2.17T + 11T^{2} \) |
| 13 | \( 1 - 0.563T + 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 - 7.02T + 19T^{2} \) |
| 31 | \( 1 + 5.22T + 31T^{2} \) |
| 37 | \( 1 + 3.21T + 37T^{2} \) |
| 41 | \( 1 + 4.83T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 6.11T + 47T^{2} \) |
| 53 | \( 1 + 8.92T + 53T^{2} \) |
| 59 | \( 1 + 6.86T + 59T^{2} \) |
| 61 | \( 1 - 1.96T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 9.69T + 71T^{2} \) |
| 73 | \( 1 - 7.14T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 6.81T + 83T^{2} \) |
| 89 | \( 1 + 5.39T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−7.52831366797165614790745023970, −6.64156970561553055716276055098, −6.24120555021403515289103046731, −5.50354324039471722144006994801, −5.06100881195325529608624352977, −3.72189897695937130673638809706, −3.51371135680516773168793724956, −2.66914426876675486925090078707, −1.41006817835324224382213855474, 0,
1.41006817835324224382213855474, 2.66914426876675486925090078707, 3.51371135680516773168793724956, 3.72189897695937130673638809706, 5.06100881195325529608624352977, 5.50354324039471722144006994801, 6.24120555021403515289103046731, 6.64156970561553055716276055098, 7.52831366797165614790745023970
