| L(s) = 1 | + 0.422·2-s − 3·3-s − 7.82·4-s + 0.947·5-s − 1.26·6-s − 7·7-s − 6.67·8-s + 9·9-s + 0.399·10-s + 29.5·11-s + 23.4·12-s + 40.6·13-s − 2.95·14-s − 2.84·15-s + 59.7·16-s + 12.0·17-s + 3.79·18-s + 19.4·19-s − 7.41·20-s + 21·21-s + 12.4·22-s + 23·23-s + 20.0·24-s − 124.·25-s + 17.1·26-s − 27·27-s + 54.7·28-s + ⋯ |
| L(s) = 1 | + 0.149·2-s − 0.577·3-s − 0.977·4-s + 0.0847·5-s − 0.0861·6-s − 0.377·7-s − 0.295·8-s + 0.333·9-s + 0.0126·10-s + 0.809·11-s + 0.564·12-s + 0.866·13-s − 0.0563·14-s − 0.0489·15-s + 0.933·16-s + 0.171·17-s + 0.0497·18-s + 0.235·19-s − 0.0828·20-s + 0.218·21-s + 0.120·22-s + 0.208·23-s + 0.170·24-s − 0.992·25-s + 0.129·26-s − 0.192·27-s + 0.369·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 0.422T + 8T^{2} \) |
| 5 | \( 1 - 0.947T + 125T^{2} \) |
| 11 | \( 1 - 29.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 12.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 19.4T + 6.85e3T^{2} \) |
| 29 | \( 1 + 190.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 189.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 240.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 171.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 420.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 200.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 375.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 339.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 260.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 532.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 482.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 60.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + 648.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.00e3T + 9.12e5T^{2} \) |
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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
−9.958681278456140148605491175152, −9.352877127602934301721720247478, −8.469800808072315885477997912561, −7.30611248851938920295168332135, −6.12961013043743942006624247415, −5.47716133929350087284131547450, −4.23333902595314629209910545042, −3.46575165864159737931887065862, −1.40804601143411956787949942402, 0,
1.40804601143411956787949942402, 3.46575165864159737931887065862, 4.23333902595314629209910545042, 5.47716133929350087284131547450, 6.12961013043743942006624247415, 7.30611248851938920295168332135, 8.469800808072315885477997912561, 9.352877127602934301721720247478, 9.958681278456140148605491175152
