| L(s) = 1 | − 3.99·2-s − 7.11·3-s − 16.0·4-s + 28.4·6-s + 173.·7-s + 191.·8-s − 192.·9-s − 87.6·11-s + 114.·12-s − 230.·13-s − 692.·14-s − 253.·16-s − 289·17-s + 768.·18-s − 1.76e3·19-s − 1.23e3·21-s + 350.·22-s + 1.15e3·23-s − 1.36e3·24-s + 920.·26-s + 3.09e3·27-s − 2.78e3·28-s + 987.·29-s + 1.68e3·31-s − 5.13e3·32-s + 623.·33-s + 1.15e3·34-s + ⋯ |
| L(s) = 1 | − 0.706·2-s − 0.456·3-s − 0.501·4-s + 0.322·6-s + 1.33·7-s + 1.06·8-s − 0.791·9-s − 0.218·11-s + 0.228·12-s − 0.378·13-s − 0.943·14-s − 0.247·16-s − 0.242·17-s + 0.559·18-s − 1.11·19-s − 0.609·21-s + 0.154·22-s + 0.456·23-s − 0.483·24-s + 0.266·26-s + 0.817·27-s − 0.670·28-s + 0.218·29-s + 0.315·31-s − 0.885·32-s + 0.0996·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 17 | \( 1 + 289T \) |
| good | 2 | \( 1 + 3.99T + 32T^{2} \) |
| 3 | \( 1 + 7.11T + 243T^{2} \) |
| 7 | \( 1 - 173.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 87.6T + 1.61e5T^{2} \) |
| 13 | \( 1 + 230.T + 3.71e5T^{2} \) |
| 19 | \( 1 + 1.76e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 987.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.30e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.23e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.62e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.35e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.54e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.64e4T + 8.58e9T^{2} \) |
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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
−9.909708771229621530843031318371, −8.835544980920556475539632976991, −8.255063715353407261540654236149, −7.44601802477651422900314989292, −6.05265198700705546634237660527, −4.97368812948814921095944645547, −4.32694867019283860338811126468, −2.46025747588138757598533111963, −1.13495619310651421413159028303, 0,
1.13495619310651421413159028303, 2.46025747588138757598533111963, 4.32694867019283860338811126468, 4.97368812948814921095944645547, 6.05265198700705546634237660527, 7.44601802477651422900314989292, 8.255063715353407261540654236149, 8.835544980920556475539632976991, 9.909708771229621530843031318371
