| L(s) = 1 | + 1.56·2-s + 0.438·4-s + 3.56·5-s + 0.561·7-s − 2.43·8-s + 5.56·10-s + 2·11-s + 0.876·14-s − 4.68·16-s + 1.56·17-s + 7.12·19-s + 1.56·20-s + 3.12·22-s − 2·23-s + 7.68·25-s + 0.246·28-s − 6.68·29-s + 2.56·31-s − 2.43·32-s + 2.43·34-s + 2·35-s + 7.56·37-s + 11.1·38-s − 8.68·40-s + 1.56·41-s + 4.56·43-s + 0.876·44-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.219·4-s + 1.59·5-s + 0.212·7-s − 0.862·8-s + 1.75·10-s + 0.603·11-s + 0.234·14-s − 1.17·16-s + 0.378·17-s + 1.63·19-s + 0.349·20-s + 0.665·22-s − 0.417·23-s + 1.53·25-s + 0.0465·28-s − 1.24·29-s + 0.460·31-s − 0.431·32-s + 0.418·34-s + 0.338·35-s + 1.24·37-s + 1.80·38-s − 1.37·40-s + 0.243·41-s + 0.695·43-s + 0.132·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.789030859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.789030859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 - 0.561T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 1.56T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 - 7.56T + 37T^{2} \) |
| 41 | \( 1 - 1.56T + 41T^{2} \) |
| 43 | \( 1 - 4.56T + 43T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 - 0.684T + 53T^{2} \) |
| 59 | \( 1 - 2.87T + 59T^{2} \) |
| 61 | \( 1 - 3.87T + 61T^{2} \) |
| 67 | \( 1 - 4.56T + 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 5.43T + 79T^{2} \) |
| 83 | \( 1 - 0.876T + 83T^{2} \) |
| 89 | \( 1 + 4.87T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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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
−9.575064592006727561361200432287, −8.932062938579776362028256459747, −7.74163278275719554611187347649, −6.65915589160477294584562458524, −5.87558146837307924857942616510, −5.44924401206447988889956578646, −4.56840936788280657444995943021, −3.49139938350712492155446945662, −2.55006818817911547437275004447, −1.36103810056341249886701136358,
1.36103810056341249886701136358, 2.55006818817911547437275004447, 3.49139938350712492155446945662, 4.56840936788280657444995943021, 5.44924401206447988889956578646, 5.87558146837307924857942616510, 6.65915589160477294584562458524, 7.74163278275719554611187347649, 8.932062938579776362028256459747, 9.575064592006727561361200432287
