| L(s) = 1 | − 4.34·2-s − 40.5·3-s − 109.·4-s − 341.·5-s + 176.·6-s + 956.·7-s + 1.03e3·8-s − 542.·9-s + 1.48e3·10-s − 3.65e3·11-s + 4.42e3·12-s + 1.15e4·13-s − 4.16e3·14-s + 1.38e4·15-s + 9.47e3·16-s + 2.36e3·17-s + 2.35e3·18-s − 9.48e3·19-s + 3.72e4·20-s − 3.87e4·21-s + 1.59e4·22-s + 1.15e4·23-s − 4.18e4·24-s + 3.87e4·25-s − 5.04e4·26-s + 1.10e5·27-s − 1.04e5·28-s + ⋯ |
| L(s) = 1 | − 0.384·2-s − 0.867·3-s − 0.852·4-s − 1.22·5-s + 0.333·6-s + 1.05·7-s + 0.712·8-s − 0.247·9-s + 0.470·10-s − 0.829·11-s + 0.739·12-s + 1.46·13-s − 0.405·14-s + 1.06·15-s + 0.578·16-s + 0.116·17-s + 0.0953·18-s − 0.317·19-s + 1.04·20-s − 0.914·21-s + 0.318·22-s + 0.198·23-s − 0.617·24-s + 0.495·25-s − 0.562·26-s + 1.08·27-s − 0.898·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5795183153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5795183153\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 2.43e4T \) |
| good | 2 | \( 1 + 4.34T + 128T^{2} \) |
| 3 | \( 1 + 40.5T + 2.18e3T^{2} \) |
| 5 | \( 1 + 341.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 956.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.65e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.15e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.36e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 9.48e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.15e4T + 3.40e9T^{2} \) |
| 31 | \( 1 + 1.84e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.10e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.24e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.23e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.77e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.50e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.04e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.02e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.82e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.34e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.66e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.10e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.18e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.60e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.54e6T + 8.07e13T^{2} \) |
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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
−15.77208300633778583626586091820, −14.38907730227103051286919468184, −12.90603154661514042316594529186, −11.42785542285497579463808651103, −10.76717249573297363636723525803, −8.656721842254161490605606315867, −7.76878337621932861643896970524, −5.48310650726764711306783976704, −4.10858097000243608815795794096, −0.70429951748339447639979373638,
0.70429951748339447639979373638, 4.10858097000243608815795794096, 5.48310650726764711306783976704, 7.76878337621932861643896970524, 8.656721842254161490605606315867, 10.76717249573297363636723525803, 11.42785542285497579463808651103, 12.90603154661514042316594529186, 14.38907730227103051286919468184, 15.77208300633778583626586091820
