L(s) = 1 | + (−0.891 + 1.54i)5-s + (2.54 + 0.727i)7-s + (2.80 + 4.86i)11-s + (3.14 + 5.43i)13-s + (−0.646 + 1.11i)17-s + (−0.559 − 0.968i)19-s + (−3.80 + 6.59i)23-s + (0.909 + 1.57i)25-s + (1.57 − 2.72i)29-s − 1.00·31-s + (−3.39 + 3.28i)35-s + (−5.96 − 10.3i)37-s + (−4.14 − 7.17i)41-s + (−2.34 + 4.06i)43-s − 1.94·47-s + ⋯ |
L(s) = 1 | + (−0.398 + 0.690i)5-s + (0.961 + 0.274i)7-s + (0.846 + 1.46i)11-s + (0.870 + 1.50i)13-s + (−0.156 + 0.271i)17-s + (−0.128 − 0.222i)19-s + (−0.794 + 1.37i)23-s + (0.181 + 0.315i)25-s + (0.292 − 0.506i)29-s − 0.180·31-s + (−0.573 + 0.554i)35-s + (−0.980 − 1.69i)37-s + (−0.646 − 1.12i)41-s + (−0.358 + 0.620i)43-s − 0.283·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.878005816\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.878005816\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.54 - 0.727i)T \) |
good | 5 | \( 1 + (0.891 - 1.54i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.80 - 4.86i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.14 - 5.43i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.646 - 1.11i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.559 + 0.968i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.80 - 6.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.57 + 2.72i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.00T + 31T^{2} \) |
| 37 | \( 1 + (5.96 + 10.3i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.14 + 7.17i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.34 - 4.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 1.94T + 47T^{2} \) |
| 53 | \( 1 + (-4.45 + 7.72i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 8.38T + 59T^{2} \) |
| 61 | \( 1 - 4.82T + 61T^{2} \) |
| 67 | \( 1 - 2.55T + 67T^{2} \) |
| 71 | \( 1 + 8.86T + 71T^{2} \) |
| 73 | \( 1 + (-5.67 + 9.83i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + (1.60 - 2.77i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.404 - 0.700i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.10 + 1.91i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.974024437815440667264068682739, −8.253267926155534021065847085011, −7.13804494269384581215918843503, −7.06503001458374396706603973970, −6.00713118286438984885307356860, −5.04190315415987498120157222124, −4.09275640524191393304014198913, −3.71201261776141301185649787476, −2.06965122266451863712809211016, −1.66272526156550039279208056807,
0.63235063279068731352036499435, 1.36687567938574967645167213477, 2.91595421218207479299642531281, 3.76376973799933623843848308496, 4.56338622449235287709749521399, 5.38128792823839302559794448181, 6.12424195496193307051917989602, 6.94872507879497623038928855117, 8.234314481125076699615828868141, 8.342287673336571741847075758979