L(s) = 1 | + 1.41·2-s − 1.73i·3-s + 2.00·4-s − 2.23i·5-s − 2.44i·6-s + 2.82·8-s − 2.99·9-s − 3.16i·10-s − 18.0·11-s − 3.46i·12-s − 18.6i·13-s − 3.87·15-s + 4.00·16-s − 1.54i·17-s − 4.24·18-s + 34.0i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.500·4-s − 0.447i·5-s − 0.408i·6-s + 0.353·8-s − 0.333·9-s − 0.316i·10-s − 1.64·11-s − 0.288i·12-s − 1.43i·13-s − 0.258·15-s + 0.250·16-s − 0.0906i·17-s − 0.235·18-s + 1.79i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1754075425\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1754075425\) |
\(L(2)\) |
|
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 | \( 1 + 1.73iT \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 18.0T + 121T^{2} \) |
| 13 | \( 1 + 18.6iT - 169T^{2} \) |
| 17 | \( 1 + 1.54iT - 289T^{2} \) |
| 19 | \( 1 - 34.0iT - 361T^{2} \) |
| 23 | \( 1 + 26.9T + 529T^{2} \) |
| 29 | \( 1 + 16.4T + 841T^{2} \) |
| 31 | \( 1 - 27.9iT - 961T^{2} \) |
| 37 | \( 1 - 51.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 37.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 63.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 28.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 0.443T + 2.80e3T^{2} \) |
| 59 | \( 1 + 74.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 105. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 12.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 45.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + 36.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 133.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 49.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 99.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 150. iT - 9.40e3T^{2} \) |
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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
−8.354329071793163272699628273288, −8.016339256100757578390958646714, −7.34205541133454697247270408347, −5.99402142414874229728799136575, −5.63084937543305616308474514401, −4.76702371160003989469181020525, −3.53068423537771859888445957655, −2.66641618426686005097644732907, −1.53301702801071070697704962607, −0.03389778743437913129245694015,
2.15260195735725743400243430695, 2.83787978858467602744694365905, 4.02686630015366453741981375668, 4.70471540100339654538997451560, 5.56598081283311027521892461169, 6.44262083049373051036179701076, 7.29890670568250570636426893572, 8.084353947373999914629836774660, 9.156189605319743216822336865956, 9.893726702973051125827823489768