| L(s) = 1 | − 2.61·2-s − 1.15·4-s + 13.5·5-s − 7·7-s + 23.9·8-s − 35.4·10-s − 24.0·11-s + 23.9·13-s + 18.3·14-s − 53.3·16-s − 79.6·17-s − 50.0·19-s − 15.6·20-s + 62.9·22-s + 151.·23-s + 58.4·25-s − 62.7·26-s + 8.10·28-s − 256.·29-s − 2.72·31-s − 51.9·32-s + 208.·34-s − 94.8·35-s + 319.·37-s + 130.·38-s + 324.·40-s − 164.·41-s + ⋯ |
| L(s) = 1 | − 0.924·2-s − 0.144·4-s + 1.21·5-s − 0.377·7-s + 1.05·8-s − 1.12·10-s − 0.659·11-s + 0.511·13-s + 0.349·14-s − 0.834·16-s − 1.13·17-s − 0.604·19-s − 0.175·20-s + 0.610·22-s + 1.37·23-s + 0.467·25-s − 0.473·26-s + 0.0546·28-s − 1.64·29-s − 0.0157·31-s − 0.287·32-s + 1.05·34-s − 0.457·35-s + 1.42·37-s + 0.558·38-s + 1.28·40-s − 0.627·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 2.61T + 8T^{2} \) |
| 5 | \( 1 - 13.5T + 125T^{2} \) |
| 11 | \( 1 + 24.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 23.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 79.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 151.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 256.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.72T + 2.97e4T^{2} \) |
| 37 | \( 1 - 319.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 164.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 422.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 100.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 194.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 576.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 43.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 509.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 894.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 14.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.53e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.77e3T + 9.12e5T^{2} \) |
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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
−9.558060495261096942553107925081, −9.291335831069693266358396980082, −8.374600039262016503220414183978, −7.32695217960962353308980824318, −6.31933545583328125087207786775, −5.36474293717675587024347076084, −4.24151790564612834136570351507, −2.59815959325628888105974289935, −1.47287285760699187193909503621, 0,
1.47287285760699187193909503621, 2.59815959325628888105974289935, 4.24151790564612834136570351507, 5.36474293717675587024347076084, 6.31933545583328125087207786775, 7.32695217960962353308980824318, 8.374600039262016503220414183978, 9.291335831069693266358396980082, 9.558060495261096942553107925081
