| L(s) = 1 | + 8.69·2-s − 18.3·3-s + 43.5·4-s + 25·5-s − 159.·6-s − 97.5·7-s + 100.·8-s + 94.7·9-s + 217.·10-s − 93.3·11-s − 800.·12-s + 2.32·13-s − 847.·14-s − 459.·15-s − 519.·16-s − 1.56e3·17-s + 824.·18-s − 1.92e3·19-s + 1.08e3·20-s + 1.79e3·21-s − 811.·22-s + 529·23-s − 1.84e3·24-s + 625·25-s + 20.1·26-s + 2.72e3·27-s − 4.24e3·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 1.17·3-s + 1.36·4-s + 0.447·5-s − 1.81·6-s − 0.752·7-s + 0.556·8-s + 0.390·9-s + 0.687·10-s − 0.232·11-s − 1.60·12-s + 0.00381·13-s − 1.15·14-s − 0.527·15-s − 0.507·16-s − 1.31·17-s + 0.599·18-s − 1.22·19-s + 0.609·20-s + 0.886·21-s − 0.357·22-s + 0.208·23-s − 0.655·24-s + 0.200·25-s + 0.00585·26-s + 0.719·27-s − 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 25T \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 8.69T + 32T^{2} \) |
| 3 | \( 1 + 18.3T + 243T^{2} \) |
| 7 | \( 1 + 97.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 93.3T + 1.61e5T^{2} \) |
| 13 | \( 1 - 2.32T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.56e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.92e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 5.90e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.27e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 692.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.77e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.65e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.65e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.19e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 972.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.60e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.43e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.23e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.09e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.25e4T + 8.58e9T^{2} \) |
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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
−12.40492942539912132336745512065, −11.31128395371870821483378539226, −10.54022088972910852867089573511, −9.002704591113394495923454603715, −6.81861759768460319659625091338, −6.14949987459376924401228667283, −5.24000703584467035316172751950, −4.08962742264289806514265637417, −2.45820382824182540510206559856, 0,
2.45820382824182540510206559856, 4.08962742264289806514265637417, 5.24000703584467035316172751950, 6.14949987459376924401228667283, 6.81861759768460319659625091338, 9.002704591113394495923454603715, 10.54022088972910852867089573511, 11.31128395371870821483378539226, 12.40492942539912132336745512065
