| L(s) = 1 | + 0.350·2-s − 21.5·3-s − 31.8·4-s − 25·5-s − 7.56·6-s − 22.3·8-s + 223.·9-s − 8.76·10-s − 193.·11-s + 688.·12-s + 691.·13-s + 539.·15-s + 1.01e3·16-s + 649.·17-s + 78.2·18-s + 696.·19-s + 796.·20-s − 67.7·22-s − 2.97e3·23-s + 483.·24-s + 625·25-s + 242.·26-s + 428.·27-s + 6.77e3·29-s + 189.·30-s + 151.·31-s + 1.07e3·32-s + ⋯ |
| L(s) = 1 | + 0.0619·2-s − 1.38·3-s − 0.996·4-s − 0.447·5-s − 0.0858·6-s − 0.123·8-s + 0.918·9-s − 0.0277·10-s − 0.481·11-s + 1.37·12-s + 1.13·13-s + 0.619·15-s + 0.988·16-s + 0.545·17-s + 0.0569·18-s + 0.442·19-s + 0.445·20-s − 0.0298·22-s − 1.17·23-s + 0.171·24-s + 0.200·25-s + 0.0702·26-s + 0.113·27-s + 1.49·29-s + 0.0383·30-s + 0.0283·31-s + 0.184·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.350T + 32T^{2} \) |
| 3 | \( 1 + 21.5T + 243T^{2} \) |
| 11 | \( 1 + 193.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 691.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 649.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 696.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.97e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 151.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.10e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.36e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.92e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.27e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.07e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.69e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.34e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.31e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.89e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.69e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.18e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.47e5T + 8.58e9T^{2} \) |
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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
−10.72011245779481724040637914562, −10.08665414621770953583725324247, −8.741877664867371947082940199638, −7.84202419990833750371016030838, −6.40010112330960362476884844430, −5.51552553048396383021758408746, −4.64021937632334924522700308637, −3.48053194199081156568177165902, −1.05822187658478752204260619945, 0,
1.05822187658478752204260619945, 3.48053194199081156568177165902, 4.64021937632334924522700308637, 5.51552553048396383021758408746, 6.40010112330960362476884844430, 7.84202419990833750371016030838, 8.741877664867371947082940199638, 10.08665414621770953583725324247, 10.72011245779481724040637914562
