| L(s) = 1 | − 3.83·2-s + 6.70·4-s + 11.3·5-s − 31.0·7-s + 4.97·8-s − 43.6·10-s − 20.9·11-s + 119.·14-s − 72.6·16-s − 114.·17-s + 45.1·19-s + 76.2·20-s + 80.3·22-s − 73.9·23-s + 4.29·25-s − 208.·28-s + 27.2·29-s − 179.·31-s + 238.·32-s + 438.·34-s − 353.·35-s − 354.·37-s − 173.·38-s + 56.5·40-s + 81.4·41-s − 256.·43-s − 140.·44-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.837·4-s + 1.01·5-s − 1.67·7-s + 0.219·8-s − 1.37·10-s − 0.574·11-s + 2.27·14-s − 1.13·16-s − 1.63·17-s + 0.545·19-s + 0.852·20-s + 0.778·22-s − 0.670·23-s + 0.0343·25-s − 1.40·28-s + 0.174·29-s − 1.04·31-s + 1.31·32-s + 2.21·34-s − 1.70·35-s − 1.57·37-s − 0.739·38-s + 0.223·40-s + 0.310·41-s − 0.907·43-s − 0.481·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3671176871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3671176871\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3.83T + 8T^{2} \) |
| 5 | \( 1 - 11.3T + 125T^{2} \) |
| 7 | \( 1 + 31.0T + 343T^{2} \) |
| 11 | \( 1 + 20.9T + 1.33e3T^{2} \) |
| 17 | \( 1 + 114.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 45.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 73.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 27.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 179.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 354.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 81.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 256.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 463.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 76.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 54.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 494.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 611.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 16.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 321.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 385.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 663.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 545.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 689.T + 9.12e5T^{2} \) |
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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
−9.218207406988330154960792060536, −8.657860724004420680831417207242, −7.52858684891949050935874043587, −6.77690460633455477622546665518, −6.16370630844972097989289962489, −5.16531054106695597097730511045, −3.82234601319730499708209246592, −2.59787598037130816996946349729, −1.81549428699197184389500544397, −0.34678200744443862293650361147,
0.34678200744443862293650361147, 1.81549428699197184389500544397, 2.59787598037130816996946349729, 3.82234601319730499708209246592, 5.16531054106695597097730511045, 6.16370630844972097989289962489, 6.77690460633455477622546665518, 7.52858684891949050935874043587, 8.657860724004420680831417207242, 9.218207406988330154960792060536
