| L(s) = 1 | + 22.1·2-s + 2.16·3-s + 364.·4-s + 122.·5-s + 47.9·6-s − 90.8·7-s + 5.23e3·8-s − 2.18e3·9-s + 2.71e3·10-s + 3.41e3·11-s + 787.·12-s − 1.12e4·13-s − 2.01e3·14-s + 265.·15-s + 6.95e4·16-s + 9.72e3·17-s − 4.84e4·18-s + 2.53e4·19-s + 4.46e4·20-s − 196.·21-s + 7.56e4·22-s − 5.27e4·23-s + 1.13e4·24-s − 6.31e4·25-s − 2.50e5·26-s − 9.45e3·27-s − 3.30e4·28-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 0.0462·3-s + 2.84·4-s + 0.438·5-s + 0.0907·6-s − 0.100·7-s + 3.61·8-s − 0.997·9-s + 0.859·10-s + 0.772·11-s + 0.131·12-s − 1.42·13-s − 0.196·14-s + 0.0202·15-s + 4.24·16-s + 0.479·17-s − 1.95·18-s + 0.847·19-s + 1.24·20-s − 0.00463·21-s + 1.51·22-s − 0.904·23-s + 0.167·24-s − 0.807·25-s − 2.79·26-s − 0.0924·27-s − 0.284·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.997004180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.997004180\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + 7.95e4T \) |
| good | 2 | \( 1 - 22.1T + 128T^{2} \) |
| 3 | \( 1 - 2.16T + 2.18e3T^{2} \) |
| 5 | \( 1 - 122.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 90.8T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.41e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.12e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 9.72e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.53e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.27e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.16e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.42e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 7.42e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.73e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 1.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.41e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.44e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.19e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.52e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.27e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.80e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.35e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.10e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.32e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.29e7T + 8.07e13T^{2} \) |
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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
−14.30769620983766743227110710375, −13.52102793473375979449285696154, −12.15161572206610553612411153889, −11.55242865958215995634596897426, −9.908769981668025279408377928051, −7.53977719545711111866302064619, −6.13447964086615433943317570100, −5.13410915226749823730529571096, −3.52553292236940281300784223832, −2.13827213695954792577595187478,
2.13827213695954792577595187478, 3.52553292236940281300784223832, 5.13410915226749823730529571096, 6.13447964086615433943317570100, 7.53977719545711111866302064619, 9.908769981668025279408377928051, 11.55242865958215995634596897426, 12.15161572206610553612411153889, 13.52102793473375979449285696154, 14.30769620983766743227110710375
