| L(s) = 1 | − 2-s − 0.484·3-s + 4-s + 0.484·6-s − 2.64·7-s − 8-s − 2.76·9-s + 2·11-s − 0.484·12-s − 0.484·13-s + 2.64·14-s + 16-s − 17-s + 2.76·18-s + 5.76·19-s + 1.28·21-s − 2·22-s + 1.35·23-s + 0.484·24-s + 0.484·26-s + 2.79·27-s − 2.64·28-s + 8.15·29-s − 2.09·31-s − 32-s − 0.969·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.279·3-s + 0.5·4-s + 0.197·6-s − 0.997·7-s − 0.353·8-s − 0.921·9-s + 0.603·11-s − 0.139·12-s − 0.134·13-s + 0.705·14-s + 0.250·16-s − 0.242·17-s + 0.651·18-s + 1.32·19-s + 0.279·21-s − 0.426·22-s + 0.283·23-s + 0.0989·24-s + 0.0950·26-s + 0.537·27-s − 0.498·28-s + 1.51·29-s − 0.376·31-s − 0.176·32-s − 0.168·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8100161726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8100161726\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 0.484T + 3T^{2} \) |
| 7 | \( 1 + 2.64T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 0.484T + 13T^{2} \) |
| 19 | \( 1 - 5.76T + 19T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 - 8.15T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 0.249T + 41T^{2} \) |
| 43 | \( 1 + 1.03T + 43T^{2} \) |
| 47 | \( 1 + 6.01T + 47T^{2} \) |
| 53 | \( 1 - 7.70T + 53T^{2} \) |
| 59 | \( 1 - 8.73T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 4.96T + 67T^{2} \) |
| 71 | \( 1 - 8.34T + 71T^{2} \) |
| 73 | \( 1 + 0.484T + 73T^{2} \) |
| 79 | \( 1 - 9.85T + 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 + 8.73T + 89T^{2} \) |
| 97 | \( 1 + 6.73T + 97T^{2} \) |
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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
−9.979778787779284311403289443652, −9.418367965298186230552988244843, −8.639688329026250080423029929078, −7.69271705286379404907775223667, −6.66147485912625691069197558185, −6.11032012823122765117793320613, −5.00796557051356582495755089928, −3.50633848011898872009317719533, −2.61253588561367422333222989470, −0.808720842577975485458497463546,
0.808720842577975485458497463546, 2.61253588561367422333222989470, 3.50633848011898872009317719533, 5.00796557051356582495755089928, 6.11032012823122765117793320613, 6.66147485912625691069197558185, 7.69271705286379404907775223667, 8.639688329026250080423029929078, 9.418367965298186230552988244843, 9.979778787779284311403289443652
