| L(s) = 1 | − 11.0i·2-s + (16.6 + 43.7i)3-s + 5.64·4-s − 185.·5-s + (483. − 184. i)6-s + 711. i·7-s − 1.47e3i·8-s + (−1.63e3 + 1.45e3i)9-s + 2.05e3i·10-s + 1.19e3·11-s + (93.8 + 246. i)12-s + 4.72e3·13-s + 7.86e3·14-s + (−3.09e3 − 8.11e3i)15-s − 1.56e4·16-s − 4.00e4·17-s + ⋯ |
| L(s) = 1 | − 0.977i·2-s + (0.355 + 0.934i)3-s + 0.0440·4-s − 0.664·5-s + (0.913 − 0.347i)6-s + 0.783i·7-s − 1.02i·8-s + (−0.746 + 0.665i)9-s + 0.649i·10-s + 0.271·11-s + (0.0156 + 0.0411i)12-s + 0.596·13-s + 0.766·14-s + (−0.236 − 0.621i)15-s − 0.953·16-s − 1.97·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.614436 + 0.820209i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.614436 + 0.820209i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-16.6 - 43.7i)T \) |
| 23 | \( 1 + (5.81e4 + 4.59e3i)T \) |
| good | 2 | \( 1 + 11.0iT - 128T^{2} \) |
| 5 | \( 1 + 185.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 711. iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 1.19e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.72e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.00e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.77e4iT - 8.93e8T^{2} \) |
| 29 | \( 1 - 1.82e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 1.27e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.56e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 2.35e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 8.00e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 9.23e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 9.46e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.27e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.65e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 9.50e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.41e5iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 2.82e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.07e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 8.08e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.97e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.73e6iT - 8.07e13T^{2} \) |
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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
−13.46026100297973176203395373362, −12.11005418025711357387673932282, −11.35057185577263682163288747244, −10.42043215101320101981944033923, −9.246537663168822202246932743836, −8.193707677555190555604943387236, −6.27642722164531019597103705539, −4.38773573138396926588209008401, −3.34498744987828416848417618587, −1.98175504257740398203737778068,
0.31550769293574494180534733508, 2.24893249432375028283234238756, 4.18363349797498323277886229740, 6.18901705399187382772228985666, 7.02539989166178667089172437099, 7.927659561055957395439151916008, 8.918878672724787701812930639729, 11.02278320113620726736300444784, 11.80834855125759683527123538548, 13.39401625392729428346949027319
