| L(s) = 1 | + 8.03·2-s + 9·3-s + 32.5·4-s − 14.1·5-s + 72.3·6-s + 244.·7-s + 4.38·8-s + 81·9-s − 113.·10-s − 627.·11-s + 292.·12-s − 93.1·13-s + 1.96e3·14-s − 127.·15-s − 1.00e3·16-s + 650.·18-s − 2.60e3·19-s − 459.·20-s + 2.20e3·21-s − 5.03e3·22-s − 4.54e3·23-s + 39.4·24-s − 2.92e3·25-s − 748.·26-s + 729·27-s + 7.95e3·28-s + 2.16e3·29-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 0.577·3-s + 1.01·4-s − 0.252·5-s + 0.819·6-s + 1.88·7-s + 0.0242·8-s + 0.333·9-s − 0.358·10-s − 1.56·11-s + 0.587·12-s − 0.152·13-s + 2.67·14-s − 0.145·15-s − 0.982·16-s + 0.473·18-s − 1.65·19-s − 0.256·20-s + 1.08·21-s − 2.21·22-s − 1.79·23-s + 0.0139·24-s − 0.936·25-s − 0.217·26-s + 0.192·27-s + 1.91·28-s + 0.477·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 8.03T + 32T^{2} \) |
| 5 | \( 1 + 14.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 244.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 627.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 93.1T + 3.71e5T^{2} \) |
| 19 | \( 1 + 2.60e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.54e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.16e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.36e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.10e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.36e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.11e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.13e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.07e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.81e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.65e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.09e5T + 8.58e9T^{2} \) |
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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
−8.586106214055919779983429264012, −8.109752518820186968593167871612, −7.38002504986725268035579029624, −6.03011599087361143309590964682, −5.20870820616165826999566680468, −4.47386693113149630768677215668, −3.84256828437806130849437674191, −2.41470194217837358146543743228, −1.96572142771402167326517059653, 0,
1.96572142771402167326517059653, 2.41470194217837358146543743228, 3.84256828437806130849437674191, 4.47386693113149630768677215668, 5.20870820616165826999566680468, 6.03011599087361143309590964682, 7.38002504986725268035579029624, 8.109752518820186968593167871612, 8.586106214055919779983429264012
