| L(s) = 1 | + 2.23·2-s − 1.23·3-s + 3.00·4-s − 2·5-s − 2.76·6-s + 7-s + 2.23·8-s − 1.47·9-s − 4.47·10-s − 11-s − 3.70·12-s + 3.23·13-s + 2.23·14-s + 2.47·15-s − 0.999·16-s − 3.23·17-s − 3.29·18-s + 6.47·19-s − 6.00·20-s − 1.23·21-s − 2.23·22-s + 2.47·23-s − 2.76·24-s − 25-s + 7.23·26-s + 5.52·27-s + 3.00·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.713·3-s + 1.50·4-s − 0.894·5-s − 1.12·6-s + 0.377·7-s + 0.790·8-s − 0.490·9-s − 1.41·10-s − 0.301·11-s − 1.07·12-s + 0.897·13-s + 0.597·14-s + 0.638·15-s − 0.249·16-s − 0.784·17-s − 0.775·18-s + 1.48·19-s − 1.34·20-s − 0.269·21-s − 0.476·22-s + 0.515·23-s − 0.564·24-s − 0.200·25-s + 1.41·26-s + 1.06·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.468292936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.468292936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 2.76T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 0.763T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−14.27550916790215652419511200225, −13.49781163498927232710566477325, −12.18910384179657484724636644400, −11.59981959396268078086031221575, −10.80098618032781986788043378165, −8.578069454461321602627873782574, −7.01529740975175196472359515240, −5.72112728486865574624740022438, −4.69824918973277729331348697866, −3.28451169009027979961022861738,
3.28451169009027979961022861738, 4.69824918973277729331348697866, 5.72112728486865574624740022438, 7.01529740975175196472359515240, 8.578069454461321602627873782574, 10.80098618032781986788043378165, 11.59981959396268078086031221575, 12.18910384179657484724636644400, 13.49781163498927232710566477325, 14.27550916790215652419511200225
