L(s) = 1 | + 5.15·2-s − 6.49·3-s + 18.5·4-s + 17.2·5-s − 33.4·6-s − 23.3·7-s + 54.3·8-s + 15.2·9-s + 88.9·10-s − 60.5·11-s − 120.·12-s + 10.9·13-s − 120.·14-s − 112.·15-s + 131.·16-s − 3.57·17-s + 78.5·18-s + 33.2·19-s + 320.·20-s + 151.·21-s − 312.·22-s + 63.7·23-s − 353.·24-s + 173.·25-s + 56.3·26-s + 76.4·27-s − 432.·28-s + ⋯ |
L(s) = 1 | + 1.82·2-s − 1.25·3-s + 2.31·4-s + 1.54·5-s − 2.27·6-s − 1.25·7-s + 2.40·8-s + 0.564·9-s + 2.81·10-s − 1.65·11-s − 2.90·12-s + 0.233·13-s − 2.29·14-s − 1.93·15-s + 2.05·16-s − 0.0509·17-s + 1.02·18-s + 0.401·19-s + 3.58·20-s + 1.57·21-s − 3.02·22-s + 0.577·23-s − 3.00·24-s + 1.38·25-s + 0.425·26-s + 0.544·27-s − 2.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.560321259\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.560321259\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + 43T \) |
good | 2 | \( 1 - 5.15T + 8T^{2} \) |
| 3 | \( 1 + 6.49T + 27T^{2} \) |
| 5 | \( 1 - 17.2T + 125T^{2} \) |
| 7 | \( 1 + 23.3T + 343T^{2} \) |
| 11 | \( 1 + 60.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 3.57T + 4.91e3T^{2} \) |
| 19 | \( 1 - 33.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 63.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 89.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 222.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 59.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 143.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 379.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 150.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 207.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 486.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 13.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 411.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 813.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 350.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.18e3T + 9.12e5T^{2} \) |
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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
−15.48801412621755453619771197275, −13.80727724237216374240032603737, −13.15987147008112891119161368636, −12.40205058679316898501127675983, −10.93586598781839085607401078505, −10.00896459683599676926309892380, −6.73845211370475354777199837234, −5.83976303890908103743692306354, −5.15330130451801237043567001255, −2.77663447143581097021597185479,
2.77663447143581097021597185479, 5.15330130451801237043567001255, 5.83976303890908103743692306354, 6.73845211370475354777199837234, 10.00896459683599676926309892380, 10.93586598781839085607401078505, 12.40205058679316898501127675983, 13.15987147008112891119161368636, 13.80727724237216374240032603737, 15.48801412621755453619771197275