| L(s) = 1 | − 4·2-s − 25.4·3-s + 16·4-s + 25·5-s + 101.·6-s + 169.·7-s − 64·8-s + 404.·9-s − 100·10-s + 240.·11-s − 407.·12-s + 9.99e2·13-s − 676.·14-s − 636.·15-s + 256·16-s + 215.·17-s − 1.61e3·18-s + 299.·19-s + 400·20-s − 4.30e3·21-s − 962.·22-s − 529·23-s + 1.62e3·24-s + 625·25-s − 3.99e3·26-s − 4.11e3·27-s + 2.70e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.63·3-s + 0.5·4-s + 0.447·5-s + 1.15·6-s + 1.30·7-s − 0.353·8-s + 1.66·9-s − 0.316·10-s + 0.599·11-s − 0.816·12-s + 1.64·13-s − 0.922·14-s − 0.730·15-s + 0.250·16-s + 0.181·17-s − 1.17·18-s + 0.190·19-s + 0.223·20-s − 2.13·21-s − 0.423·22-s − 0.208·23-s + 0.577·24-s + 0.200·25-s − 1.16·26-s − 1.08·27-s + 0.652·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.197314992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.197314992\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 - 25T \) |
| 23 | \( 1 + 529T \) |
| good | 3 | \( 1 + 25.4T + 243T^{2} \) |
| 7 | \( 1 - 169.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 240.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 9.99e2T + 3.71e5T^{2} \) |
| 17 | \( 1 - 215.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 299.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 7.68e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 588.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.65e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.43e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.90e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 460.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.60e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.83e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.01e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.69e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.94e4T + 8.58e9T^{2} \) |
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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
−11.16739325567448768618445593554, −10.75160085572928301793300792945, −9.519762239466829385129376461501, −8.407448651185556278288537461164, −7.22894989982321654412364158294, −6.08808639989125399494185973367, −5.47387213856618898813527198724, −4.10108576905146034407063892321, −1.68254803256294991940398988542, −0.869442933108639570102142649408,
0.869442933108639570102142649408, 1.68254803256294991940398988542, 4.10108576905146034407063892321, 5.47387213856618898813527198724, 6.08808639989125399494185973367, 7.22894989982321654412364158294, 8.407448651185556278288537461164, 9.519762239466829385129376461501, 10.75160085572928301793300792945, 11.16739325567448768618445593554
