| L(s) = 1 | + 3.15·2-s + 9·3-s − 22.0·4-s + 28.4·6-s + 35.7·7-s − 170.·8-s + 81·9-s − 121·11-s − 198.·12-s − 128.·13-s + 112.·14-s + 165.·16-s + 233.·17-s + 255.·18-s − 533.·19-s + 321.·21-s − 382.·22-s + 4.30e3·23-s − 1.53e3·24-s − 406.·26-s + 729·27-s − 786.·28-s + 7.05e3·29-s − 7.48e3·31-s + 5.98e3·32-s − 1.08e3·33-s + 736.·34-s + ⋯ |
| L(s) = 1 | + 0.558·2-s + 0.577·3-s − 0.688·4-s + 0.322·6-s + 0.275·7-s − 0.942·8-s + 0.333·9-s − 0.301·11-s − 0.397·12-s − 0.211·13-s + 0.153·14-s + 0.161·16-s + 0.195·17-s + 0.186·18-s − 0.339·19-s + 0.159·21-s − 0.168·22-s + 1.69·23-s − 0.544·24-s − 0.118·26-s + 0.192·27-s − 0.189·28-s + 1.55·29-s − 1.39·31-s + 1.03·32-s − 0.174·33-s + 0.109·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 3.15T + 32T^{2} \) |
| 7 | \( 1 - 35.7T + 1.68e4T^{2} \) |
| 13 | \( 1 + 128.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 233.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 533.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.30e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.05e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.48e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.77e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.63e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.91e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.92e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.52e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.93e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.64e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.40e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.53e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.15e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.17e4T + 8.58e9T^{2} \) |
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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
−8.952966412219039470454523200574, −8.366082566382833618214982978283, −7.39153297051628183105516105988, −6.37346036141166295689242814935, −5.17955425269535677127102386707, −4.65400186569060136354440978662, −3.51948244392430793638428746764, −2.76196365381324179487364304947, −1.34832759319246584770861638307, 0,
1.34832759319246584770861638307, 2.76196365381324179487364304947, 3.51948244392430793638428746764, 4.65400186569060136354440978662, 5.17955425269535677127102386707, 6.37346036141166295689242814935, 7.39153297051628183105516105988, 8.366082566382833618214982978283, 8.952966412219039470454523200574
