| L(s) = 1 | + 4.20·2-s − 3·3-s + 9.65·4-s − 12.6·6-s − 15.3·7-s + 6.96·8-s + 9·9-s − 11·11-s − 28.9·12-s − 24.4·13-s − 64.6·14-s − 48.0·16-s + 54.9·17-s + 37.8·18-s + 119.·19-s + 46.1·21-s − 46.2·22-s + 191.·23-s − 20.8·24-s − 102.·26-s − 27·27-s − 148.·28-s + 225.·29-s + 303.·31-s − 257.·32-s + 33·33-s + 230.·34-s + ⋯ |
| L(s) = 1 | + 1.48·2-s − 0.577·3-s + 1.20·4-s − 0.857·6-s − 0.830·7-s + 0.307·8-s + 0.333·9-s − 0.301·11-s − 0.696·12-s − 0.522·13-s − 1.23·14-s − 0.750·16-s + 0.783·17-s + 0.495·18-s + 1.44·19-s + 0.479·21-s − 0.447·22-s + 1.73·23-s − 0.177·24-s − 0.776·26-s − 0.192·27-s − 1.00·28-s + 1.44·29-s + 1.76·31-s − 1.42·32-s + 0.174·33-s + 1.16·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.441087702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.441087702\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 4.20T + 8T^{2} \) |
| 7 | \( 1 + 15.3T + 343T^{2} \) |
| 13 | \( 1 + 24.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 225.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 303.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 109.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 348.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 306.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 216.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 692.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 152.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 62.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 554.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 122.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 476.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 913.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.60e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 498.T + 9.12e5T^{2} \) |
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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
−9.976716152930629783257912520260, −9.218425930739907523438975006453, −7.78638550099056245975125106608, −6.81113123966579368842072365721, −6.19547146977596008063366292741, −5.17249283676517564982794944095, −4.72557065167434015558778232018, −3.34813854355423216941823578883, −2.79072913182173734402393813232, −0.855915469735104951197775373955,
0.855915469735104951197775373955, 2.79072913182173734402393813232, 3.34813854355423216941823578883, 4.72557065167434015558778232018, 5.17249283676517564982794944095, 6.19547146977596008063366292741, 6.81113123966579368842072365721, 7.78638550099056245975125106608, 9.218425930739907523438975006453, 9.976716152930629783257912520260
