| L(s) = 1 | + 1.64·2-s − 3-s + 0.707·4-s + 3.84·5-s − 1.64·6-s + 3.11·7-s − 2.12·8-s + 9-s + 6.32·10-s + 11-s − 0.707·12-s + 5.12·14-s − 3.84·15-s − 4.91·16-s − 0.804·17-s + 1.64·18-s − 6.44·19-s + 2.72·20-s − 3.11·21-s + 1.64·22-s + 7.57·23-s + 2.12·24-s + 9.77·25-s − 27-s + 2.20·28-s + 4.79·29-s − 6.32·30-s + ⋯ |
| L(s) = 1 | + 1.16·2-s − 0.577·3-s + 0.353·4-s + 1.71·5-s − 0.671·6-s + 1.17·7-s − 0.751·8-s + 0.333·9-s + 2.00·10-s + 0.301·11-s − 0.204·12-s + 1.36·14-s − 0.992·15-s − 1.22·16-s − 0.195·17-s + 0.387·18-s − 1.47·19-s + 0.608·20-s − 0.679·21-s + 0.350·22-s + 1.58·23-s + 0.434·24-s + 1.95·25-s − 0.192·27-s + 0.416·28-s + 0.890·29-s − 1.15·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.590658857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.590658857\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.64T + 2T^{2} \) |
| 5 | \( 1 - 3.84T + 5T^{2} \) |
| 7 | \( 1 - 3.11T + 7T^{2} \) |
| 17 | \( 1 + 0.804T + 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 - 7.57T + 23T^{2} \) |
| 29 | \( 1 - 4.79T + 29T^{2} \) |
| 31 | \( 1 + 3.74T + 31T^{2} \) |
| 37 | \( 1 - 0.628T + 37T^{2} \) |
| 41 | \( 1 - 8.72T + 41T^{2} \) |
| 43 | \( 1 + 5.27T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 4.85T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 1.77T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 0.287T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 0.659T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.224413694984358211593952836531, −6.90576909344749517795309285847, −6.53888098783666942004951919182, −5.69368924485344260447394841464, −5.31719612426226642211597647146, −4.66480193477747916685047328856, −4.03692718609175935875040999699, −2.70756270647952052176956323526, −2.07444153008320294977922157847, −1.05367312038561477566295379910,
1.05367312038561477566295379910, 2.07444153008320294977922157847, 2.70756270647952052176956323526, 4.03692718609175935875040999699, 4.66480193477747916685047328856, 5.31719612426226642211597647146, 5.69368924485344260447394841464, 6.53888098783666942004951919182, 6.90576909344749517795309285847, 8.224413694984358211593952836531
