| L(s) = 1 | − 15.2·2-s + 103.·4-s + 537.·5-s + 1.47e3·7-s + 371.·8-s − 8.18e3·10-s + 3.35e3·11-s − 1.24e4·13-s − 2.23e4·14-s − 1.89e4·16-s − 2.07e4·17-s − 1.06e3·19-s + 5.57e4·20-s − 5.10e4·22-s + 1.78e4·23-s + 2.11e5·25-s + 1.88e5·26-s + 1.52e5·28-s − 9.99e4·29-s − 3.01e4·31-s + 2.40e5·32-s + 3.16e5·34-s + 7.91e5·35-s − 1.74e5·37-s + 1.62e4·38-s + 1.99e5·40-s + 4.50e4·41-s + ⋯ |
| L(s) = 1 | − 1.34·2-s + 0.809·4-s + 1.92·5-s + 1.62·7-s + 0.256·8-s − 2.58·10-s + 0.759·11-s − 1.56·13-s − 2.18·14-s − 1.15·16-s − 1.02·17-s − 0.0356·19-s + 1.55·20-s − 1.02·22-s + 0.305·23-s + 2.70·25-s + 2.10·26-s + 1.31·28-s − 0.760·29-s − 0.181·31-s + 1.29·32-s + 1.37·34-s + 3.12·35-s − 0.566·37-s + 0.0480·38-s + 0.493·40-s + 0.102·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.995290134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995290134\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 - 7.95e4T \) |
| good | 2 | \( 1 + 15.2T + 128T^{2} \) |
| 5 | \( 1 - 537.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.47e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.35e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.24e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.07e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.06e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.78e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.99e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.01e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.74e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.50e4T + 1.94e11T^{2} \) |
| 47 | \( 1 - 3.19e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.27e4T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.67e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.37e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.70e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.04e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 8.62e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.18e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.05e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.05e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.49e5T + 8.07e13T^{2} \) |
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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
−9.934257681879740993865906973631, −9.224765709042772399035450403184, −8.649448808285871407579160419028, −7.48149581807527149889388280711, −6.65882941808174882772030887427, −5.33020414016580437805616521707, −4.61934477772635288165314097012, −2.18189880813998067097648271025, −1.91684364845485277098863392613, −0.834295631221654714204793971798,
0.834295631221654714204793971798, 1.91684364845485277098863392613, 2.18189880813998067097648271025, 4.61934477772635288165314097012, 5.33020414016580437805616521707, 6.65882941808174882772030887427, 7.48149581807527149889388280711, 8.649448808285871407579160419028, 9.224765709042772399035450403184, 9.934257681879740993865906973631
