| L(s) = 1 | + 4·2-s + 16·4-s − 66.0·5-s + 114.·7-s + 64·8-s − 264.·10-s − 385.·11-s + 1.03e3·13-s + 456.·14-s + 256·16-s + 959.·17-s − 464.·19-s − 1.05e3·20-s − 1.54e3·22-s + 2.30e3·23-s + 1.24e3·25-s + 4.13e3·26-s + 1.82e3·28-s + 7.09e3·29-s + 7.76e3·31-s + 1.02e3·32-s + 3.83e3·34-s − 7.54e3·35-s + 9.31e3·37-s − 1.85e3·38-s − 4.22e3·40-s − 1.33e4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.18·5-s + 0.880·7-s + 0.353·8-s − 0.835·10-s − 0.960·11-s + 1.69·13-s + 0.622·14-s + 0.250·16-s + 0.804·17-s − 0.295·19-s − 0.591·20-s − 0.679·22-s + 0.907·23-s + 0.397·25-s + 1.19·26-s + 0.440·28-s + 1.56·29-s + 1.45·31-s + 0.176·32-s + 0.569·34-s − 1.04·35-s + 1.11·37-s − 0.208·38-s − 0.417·40-s − 1.23·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.868443724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.868443724\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 66.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 114.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 385.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 959.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 464.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.30e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.09e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.31e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.33e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.11e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.49e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.44e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.35e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 970.T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.60e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.38e4T + 8.58e9T^{2} \) |
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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
−11.85299434083253530670357228227, −11.23709172135670409596115891425, −10.34372309153675252168838391877, −8.359081598332945336565772801503, −7.971539383238943188507633334315, −6.57084890434211105508921864475, −5.19493340218067527117145220638, −4.17102231297193029096594964119, −2.98920319023737163418652930024, −1.06093840391283193035008657608,
1.06093840391283193035008657608, 2.98920319023737163418652930024, 4.17102231297193029096594964119, 5.19493340218067527117145220638, 6.57084890434211105508921864475, 7.971539383238943188507633334315, 8.359081598332945336565772801503, 10.34372309153675252168838391877, 11.23709172135670409596115891425, 11.85299434083253530670357228227
