| L(s) = 1 | − 42.3·2-s + 109.·3-s + 1.28e3·4-s − 2.49e3·5-s − 4.65e3·6-s + 2.87e3·7-s − 3.27e4·8-s − 7.64e3·9-s + 1.05e5·10-s + 3.65e4·11-s + 1.41e5·12-s + 6.99e4·13-s − 1.21e5·14-s − 2.74e5·15-s + 7.32e5·16-s + 8.35e4·17-s + 3.23e5·18-s + 6.40e5·19-s − 3.21e6·20-s + 3.15e5·21-s − 1.55e6·22-s + 2.09e6·23-s − 3.59e6·24-s + 4.28e6·25-s − 2.96e6·26-s − 2.99e6·27-s + 3.69e6·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 0.782·3-s + 2.51·4-s − 1.78·5-s − 1.46·6-s + 0.452·7-s − 2.83·8-s − 0.388·9-s + 3.34·10-s + 0.753·11-s + 1.96·12-s + 0.678·13-s − 0.847·14-s − 1.39·15-s + 2.79·16-s + 0.242·17-s + 0.727·18-s + 1.12·19-s − 4.48·20-s + 0.353·21-s − 1.41·22-s + 1.55·23-s − 2.21·24-s + 2.19·25-s − 1.27·26-s − 1.08·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.7106007450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7106007450\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - 8.35e4T \) |
| good | 2 | \( 1 + 42.3T + 512T^{2} \) |
| 3 | \( 1 - 109.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.49e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.87e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.65e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 6.99e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 6.40e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.09e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.99e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.61e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.47e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.69e5T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.50e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.55e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.03e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.79e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.79e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 8.26e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.03e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.43e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.90e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.47e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.95e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.49e8T + 7.60e17T^{2} \) |
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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
−16.79051241290812732677108853013, −15.72410159685632248549507383900, −14.70391314353486816083546816534, −11.82803901583009412551996773756, −11.04999628487850933439795979740, −9.051113358499463601589838765392, −8.226234014675006777366193697748, −7.21342054371703754541035211976, −3.24490865549226043494539667760, −0.915894036795500598442491894205,
0.915894036795500598442491894205, 3.24490865549226043494539667760, 7.21342054371703754541035211976, 8.226234014675006777366193697748, 9.051113358499463601589838765392, 11.04999628487850933439795979740, 11.82803901583009412551996773756, 14.70391314353486816083546816534, 15.72410159685632248549507383900, 16.79051241290812732677108853013
