| L(s) = 1 | − 1.41·2-s + 2.00·4-s − 8.12i·5-s + (−4 + 5.74i)7-s − 2.82·8-s + 11.4i·10-s − 12.7·11-s − 22.9i·13-s + (5.65 − 8.12i)14-s + 4.00·16-s − 8.12i·17-s − 11.4i·19-s − 16.2i·20-s + 18·22-s − 21.2·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s − 1.62i·5-s + (−0.571 + 0.820i)7-s − 0.353·8-s + 1.14i·10-s − 1.15·11-s − 1.76i·13-s + (0.404 − 0.580i)14-s + 0.250·16-s − 0.477i·17-s − 0.604i·19-s − 0.812i·20-s + 0.818·22-s − 0.922·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.299133 - 0.572797i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.299133 - 0.572797i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (4 - 5.74i)T \) |
| good | 5 | \( 1 + 8.12iT - 25T^{2} \) |
| 11 | \( 1 + 12.7T + 121T^{2} \) |
| 13 | \( 1 + 22.9iT - 169T^{2} \) |
| 17 | \( 1 + 8.12iT - 289T^{2} \) |
| 19 | \( 1 + 11.4iT - 361T^{2} \) |
| 23 | \( 1 + 21.2T + 529T^{2} \) |
| 29 | \( 1 - 33.9T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 16T + 1.36e3T^{2} \) |
| 41 | \( 1 - 56.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 52T + 1.84e3T^{2} \) |
| 47 | \( 1 + 32.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 16.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 32.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 22.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 52T + 4.48e3T^{2} \) |
| 71 | \( 1 - 89.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 45.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 104T + 6.24e3T^{2} \) |
| 83 | \( 1 + 162. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 73.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 91.9iT - 9.40e3T^{2} \) |
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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
−12.71466534974326276522611451924, −11.93966068927524123686808573672, −10.43177324090879846875909008429, −9.506872784552101045731784993452, −8.499190322599170232933594039604, −7.82160738204052593435463160007, −5.90050167021205748787454122921, −4.99130214645521398519526447503, −2.72946618301696376772331510698, −0.53443547214887323211108638913,
2.39419391649487292826104733910, 3.87407045650152377376382153061, 6.22724533109283352828902654850, 7.00348277058581787091935621079, 7.950292834003885802741715748291, 9.578053506161439439882526896306, 10.43349170247942498041528426992, 11.00557098193655280625725281262, 12.26073158115589482260013297710, 13.79687524443062814219271645294
