L(s) = 1 | + (1.75 + 0.950i)2-s + (2.19 + 3.34i)4-s − 3.98·7-s + (0.677 + 7.97i)8-s + 8.39i·11-s − 9.77i·13-s + (−7.02 − 3.79i)14-s + (−6.38 + 14.6i)16-s + 12.1i·17-s + 17.3i·19-s + (−7.97 + 14.7i)22-s + 2.97·23-s + (9.28 − 17.1i)26-s + (−8.74 − 13.3i)28-s − 51.6·29-s + ⋯ |
L(s) = 1 | + (0.879 + 0.475i)2-s + (0.548 + 0.836i)4-s − 0.569·7-s + (0.0847 + 0.996i)8-s + 0.763i·11-s − 0.751i·13-s + (−0.501 − 0.270i)14-s + (−0.399 + 0.916i)16-s + 0.714i·17-s + 0.914i·19-s + (−0.362 + 0.671i)22-s + 0.129·23-s + (0.357 − 0.661i)26-s + (−0.312 − 0.476i)28-s − 1.78·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 - 0.502i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.194035176\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.194035176\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.75 - 0.950i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.98T + 49T^{2} \) |
| 11 | \( 1 - 8.39iT - 121T^{2} \) |
| 13 | \( 1 + 9.77iT - 169T^{2} \) |
| 17 | \( 1 - 12.1iT - 289T^{2} \) |
| 19 | \( 1 - 17.3iT - 361T^{2} \) |
| 23 | \( 1 - 2.97T + 529T^{2} \) |
| 29 | \( 1 + 51.6T + 841T^{2} \) |
| 31 | \( 1 - 30.7iT - 961T^{2} \) |
| 37 | \( 1 - 19.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 42.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 62.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 39.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 21.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 50.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 70.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 94.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 132. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 77.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 48.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 140.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 18.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 15iT - 9.40e3T^{2} \) |
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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
−10.33982124981779040360964290719, −9.428250518013891117777718088630, −8.307415475465772092485463708284, −7.59586905411729217839646971720, −6.72168018737636611716510874570, −5.88951461654854172915302051378, −5.08235718448000013491663439094, −3.95303638427230325918752274932, −3.17481134878556586150558560191, −1.85156188436918510159449443728,
0.49205971198528419266698726809, 2.08076723072649294796933355591, 3.14670090219568117052588913618, 4.03528552444971667333323726896, 5.07805754626819646100849490657, 5.98082136218670524477244737461, 6.76583694143477806512481914572, 7.63909162384342716328263450896, 9.154425823758747349891236213161, 9.485772414180459373425670590083