| L(s) = 1 | + 1.65·2-s − 9·3-s − 29.2·4-s + 59.9·5-s − 14.9·6-s − 87.2·7-s − 101.·8-s + 81·9-s + 99.4·10-s + 655.·11-s + 263.·12-s + 139.·13-s − 144.·14-s − 539.·15-s + 767.·16-s − 1.51e3·17-s + 134.·18-s + 393.·19-s − 1.75e3·20-s + 785.·21-s + 1.08e3·22-s − 4.74e3·23-s + 914.·24-s + 466.·25-s + 232.·26-s − 729·27-s + 2.55e3·28-s + ⋯ |
| L(s) = 1 | + 0.293·2-s − 0.577·3-s − 0.913·4-s + 1.07·5-s − 0.169·6-s − 0.672·7-s − 0.561·8-s + 0.333·9-s + 0.314·10-s + 1.63·11-s + 0.527·12-s + 0.229·13-s − 0.197·14-s − 0.618·15-s + 0.749·16-s − 1.27·17-s + 0.0977·18-s + 0.250·19-s − 0.979·20-s + 0.388·21-s + 0.479·22-s − 1.87·23-s + 0.324·24-s + 0.149·25-s + 0.0673·26-s − 0.192·27-s + 0.615·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 - 1.65T + 32T^{2} \) |
| 5 | \( 1 - 59.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 87.2T + 1.68e4T^{2} \) |
| 11 | \( 1 - 655.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 139.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.51e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 393.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.74e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.80e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.71e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.39e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.93e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.57e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.56e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 1.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.30e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.49e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.09e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.36e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.39e5T + 8.58e9T^{2} \) |
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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
−11.48029239219723659230717217167, −9.959753471635539700121014989088, −9.549167733143926208900895008114, −8.490023872608393022492700532429, −6.50950284470743050813168071340, −6.09684394869935379223898342811, −4.73681531580150838531449960476, −3.63027831436270921371114883529, −1.64825052274665905681361866588, 0,
1.64825052274665905681361866588, 3.63027831436270921371114883529, 4.73681531580150838531449960476, 6.09684394869935379223898342811, 6.50950284470743050813168071340, 8.490023872608393022492700532429, 9.549167733143926208900895008114, 9.959753471635539700121014989088, 11.48029239219723659230717217167
