| L(s) = 1 | − 2.01·2-s + 2.05·4-s − 3.97·5-s + 1.72·7-s − 0.101·8-s + 7.99·10-s + 3.49·11-s − 0.572·13-s − 3.46·14-s − 3.89·16-s − 4.97·17-s + 8.12·19-s − 8.14·20-s − 7.03·22-s + 6.23·23-s + 10.7·25-s + 1.15·26-s + 3.52·28-s + 5.36·29-s + 10.6·31-s + 8.04·32-s + 10.0·34-s − 6.83·35-s − 5.15·37-s − 16.3·38-s + 0.402·40-s + 2.03·41-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 1.02·4-s − 1.77·5-s + 0.650·7-s − 0.0357·8-s + 2.52·10-s + 1.05·11-s − 0.158·13-s − 0.925·14-s − 0.974·16-s − 1.20·17-s + 1.86·19-s − 1.82·20-s − 1.49·22-s + 1.29·23-s + 2.15·25-s + 0.225·26-s + 0.666·28-s + 0.997·29-s + 1.90·31-s + 1.42·32-s + 1.71·34-s − 1.15·35-s − 0.848·37-s − 2.65·38-s + 0.0635·40-s + 0.318·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7984490657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7984490657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 2 | \( 1 + 2.01T + 2T^{2} \) |
| 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 - 1.72T + 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 + 0.572T + 13T^{2} \) |
| 17 | \( 1 + 4.97T + 17T^{2} \) |
| 19 | \( 1 - 8.12T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 - 5.36T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 + 5.15T + 37T^{2} \) |
| 41 | \( 1 - 2.03T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 4.85T + 47T^{2} \) |
| 53 | \( 1 + 14.5T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 + 1.97T + 67T^{2} \) |
| 71 | \( 1 - 5.81T + 71T^{2} \) |
| 73 | \( 1 + 4.38T + 73T^{2} \) |
| 79 | \( 1 + 0.158T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 - 2.84T + 89T^{2} \) |
| 97 | \( 1 - 8.72T + 97T^{2} \) |
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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
−8.369361721345010714653504627350, −7.42525885108270588025699699057, −7.13114477721084115004314299635, −6.41050564692259536046480155347, −4.80506613522820811067174790803, −4.60406346882958366645444494370, −3.57280866973961968889426343210, −2.68364717607848718649689626957, −1.27062964537222369713680195554, −0.69336586503183956584835980396,
0.69336586503183956584835980396, 1.27062964537222369713680195554, 2.68364717607848718649689626957, 3.57280866973961968889426343210, 4.60406346882958366645444494370, 4.80506613522820811067174790803, 6.41050564692259536046480155347, 7.13114477721084115004314299635, 7.42525885108270588025699699057, 8.369361721345010714653504627350
