| L(s) = 1 | + 3.74·2-s + 49.4·3-s − 113.·4-s + 80.0·5-s + 185.·6-s − 906.·8-s + 260.·9-s + 299.·10-s + 1.33e3·11-s − 5.63e3·12-s − 4.18e3·13-s + 3.96e3·15-s + 1.11e4·16-s + 3.26e4·17-s + 977.·18-s + 4.07e4·19-s − 9.12e3·20-s + 4.98e3·22-s + 2.48e3·23-s − 4.48e4·24-s − 7.17e4·25-s − 1.56e4·26-s − 9.52e4·27-s + 689.·29-s + 1.48e4·30-s − 1.27e5·31-s + 1.57e5·32-s + ⋯ |
| L(s) = 1 | + 0.331·2-s + 1.05·3-s − 0.890·4-s + 0.286·5-s + 0.350·6-s − 0.625·8-s + 0.119·9-s + 0.0948·10-s + 0.301·11-s − 0.941·12-s − 0.528·13-s + 0.303·15-s + 0.683·16-s + 1.61·17-s + 0.0394·18-s + 1.36·19-s − 0.255·20-s + 0.0998·22-s + 0.0426·23-s − 0.662·24-s − 0.917·25-s − 0.174·26-s − 0.931·27-s + 0.00524·29-s + 0.100·30-s − 0.771·31-s + 0.852·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.189834824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.189834824\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - 1.33e3T \) |
| good | 2 | \( 1 - 3.74T + 128T^{2} \) |
| 3 | \( 1 - 49.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 80.0T + 7.81e4T^{2} \) |
| 13 | \( 1 + 4.18e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.26e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.07e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.48e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 689.T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.27e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.67e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.91e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.36e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.13e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.32e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.36e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.23e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.12e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.93e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.06e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.40e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.49e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.17e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.32e6T + 8.07e13T^{2} \) |
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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
−9.667873454025425281532368433400, −8.828237609751519089973240391609, −8.047795460706898228509215458453, −7.22391136379552327495543396142, −5.68673587725560420141671102358, −5.13954196019171425338311162307, −3.67517481230392778620551500790, −3.29102974102346022406622071841, −1.97826826757972134575180937062, −0.70329710436939530993639656326,
0.70329710436939530993639656326, 1.97826826757972134575180937062, 3.29102974102346022406622071841, 3.67517481230392778620551500790, 5.13954196019171425338311162307, 5.68673587725560420141671102358, 7.22391136379552327495543396142, 8.047795460706898228509215458453, 8.828237609751519089973240391609, 9.667873454025425281532368433400
