| L(s) = 1 | − 2·2-s + 9·3-s − 28·4-s − 11·5-s − 18·6-s + 120·8-s + 81·9-s + 22·10-s + 269·11-s − 252·12-s + 308·13-s − 99·15-s + 656·16-s − 1.89e3·17-s − 162·18-s + 164·19-s + 308·20-s − 538·22-s − 3.26e3·23-s + 1.08e3·24-s − 3.00e3·25-s − 616·26-s + 729·27-s + 2.41e3·29-s + 198·30-s − 2.84e3·31-s − 5.15e3·32-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 0.577·3-s − 7/8·4-s − 0.196·5-s − 0.204·6-s + 0.662·8-s + 1/3·9-s + 0.0695·10-s + 0.670·11-s − 0.505·12-s + 0.505·13-s − 0.113·15-s + 0.640·16-s − 1.59·17-s − 0.117·18-s + 0.104·19-s + 0.172·20-s − 0.236·22-s − 1.28·23-s + 0.382·24-s − 0.961·25-s − 0.178·26-s + 0.192·27-s + 0.533·29-s + 0.0401·30-s − 0.530·31-s − 0.889·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p T + p^{5} T^{2} \) |
| 5 | \( 1 + 11 T + p^{5} T^{2} \) |
| 11 | \( 1 - 269 T + p^{5} T^{2} \) |
| 13 | \( 1 - 308 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1896 T + p^{5} T^{2} \) |
| 19 | \( 1 - 164 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3264 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2417 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2841 T + p^{5} T^{2} \) |
| 37 | \( 1 + 11328 T + p^{5} T^{2} \) |
| 41 | \( 1 - 16856 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7894 T + p^{5} T^{2} \) |
| 47 | \( 1 + 21102 T + p^{5} T^{2} \) |
| 53 | \( 1 + 29691 T + p^{5} T^{2} \) |
| 59 | \( 1 - 8163 T + p^{5} T^{2} \) |
| 61 | \( 1 + 15166 T + p^{5} T^{2} \) |
| 67 | \( 1 + 32078 T + p^{5} T^{2} \) |
| 71 | \( 1 + 38274 T + p^{5} T^{2} \) |
| 73 | \( 1 + 34866 T + p^{5} T^{2} \) |
| 79 | \( 1 - 13529 T + p^{5} T^{2} \) |
| 83 | \( 1 - 68103 T + p^{5} T^{2} \) |
| 89 | \( 1 - 114922 T + p^{5} T^{2} \) |
| 97 | \( 1 + 154959 T + p^{5} T^{2} \) |
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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
−11.60754729030854710304653799187, −10.39250579201504952137758231150, −9.337048093757830361067634944860, −8.640977851517739072351690391949, −7.68726746767675137118186534298, −6.27454797421581038594752510745, −4.58963379108487618676726613774, −3.65820207151287439259416916515, −1.73540052855481851507944316169, 0,
1.73540052855481851507944316169, 3.65820207151287439259416916515, 4.58963379108487618676726613774, 6.27454797421581038594752510745, 7.68726746767675137118186534298, 8.640977851517739072351690391949, 9.337048093757830361067634944860, 10.39250579201504952137758231150, 11.60754729030854710304653799187
