| L(s) = 1 | + 2.49·2-s + 1.93·3-s + 4.20·4-s + 5-s + 4.82·6-s − 2.50·7-s + 5.50·8-s + 0.745·9-s + 2.49·10-s + 2.95·11-s + 8.14·12-s + 1.37·13-s − 6.23·14-s + 1.93·15-s + 5.29·16-s + 5.45·17-s + 1.85·18-s − 5.04·19-s + 4.20·20-s − 4.84·21-s + 7.37·22-s + 3.29·23-s + 10.6·24-s + 25-s + 3.41·26-s − 4.36·27-s − 10.5·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 1.11·3-s + 2.10·4-s + 0.447·5-s + 1.96·6-s − 0.946·7-s + 1.94·8-s + 0.248·9-s + 0.787·10-s + 0.891·11-s + 2.35·12-s + 0.380·13-s − 1.66·14-s + 0.499·15-s + 1.32·16-s + 1.32·17-s + 0.437·18-s − 1.15·19-s + 0.941·20-s − 1.05·21-s + 1.57·22-s + 0.687·23-s + 2.17·24-s + 0.200·25-s + 0.669·26-s − 0.839·27-s − 1.99·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.391724787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.391724787\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 3 | \( 1 - 1.93T + 3T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 11 | \( 1 - 2.95T + 11T^{2} \) |
| 13 | \( 1 - 1.37T + 13T^{2} \) |
| 17 | \( 1 - 5.45T + 17T^{2} \) |
| 19 | \( 1 + 5.04T + 19T^{2} \) |
| 23 | \( 1 - 3.29T + 23T^{2} \) |
| 29 | \( 1 + 8.20T + 29T^{2} \) |
| 31 | \( 1 + 1.09T + 31T^{2} \) |
| 37 | \( 1 - 5.86T + 37T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 43 | \( 1 + 7.06T + 43T^{2} \) |
| 47 | \( 1 + 5.52T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 - 5.02T + 59T^{2} \) |
| 61 | \( 1 + 6.66T + 61T^{2} \) |
| 67 | \( 1 - 7.73T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 + 1.57T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + 9.74T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 9.23T + 97T^{2} \) |
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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
−9.326186591758468191412196878882, −8.271753711436684795076737322327, −7.36657719517828113813462665012, −6.42544249195751733083432013566, −6.02913550633448826475291471949, −5.05799226694351136826925179441, −3.89008000430940935696513626911, −3.46769506573334265420666326249, −2.70737361657445093627958151286, −1.71308633081337058894201292684,
1.71308633081337058894201292684, 2.70737361657445093627958151286, 3.46769506573334265420666326249, 3.89008000430940935696513626911, 5.05799226694351136826925179441, 6.02913550633448826475291471949, 6.42544249195751733083432013566, 7.36657719517828113813462665012, 8.271753711436684795076737322327, 9.326186591758468191412196878882
