| L(s) = 1 | + (1.58 + 0.916i)2-s + (0.822 + 2.88i)3-s + (−0.318 − 0.551i)4-s + (−1.93 + 1.11i)5-s + (−1.33 + 5.33i)6-s + (3.23 − 5.60i)7-s − 8.50i·8-s + (−7.64 + 4.74i)9-s − 4.10·10-s + (9.24 + 5.33i)11-s + (1.32 − 1.37i)12-s + (−5.53 − 9.58i)13-s + (10.2 − 5.93i)14-s + (−4.81 − 4.66i)15-s + (6.52 − 11.2i)16-s + 12.6i·17-s + ⋯ |
| L(s) = 1 | + (0.794 + 0.458i)2-s + (0.274 + 0.961i)3-s + (−0.0795 − 0.137i)4-s + (−0.387 + 0.223i)5-s + (−0.223 + 0.889i)6-s + (0.461 − 0.800i)7-s − 1.06i·8-s + (−0.849 + 0.527i)9-s − 0.410·10-s + (0.840 + 0.485i)11-s + (0.110 − 0.114i)12-s + (−0.425 − 0.737i)13-s + (0.733 − 0.423i)14-s + (−0.321 − 0.311i)15-s + (0.407 − 0.706i)16-s + 0.743i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.786i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.618 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.38722 + 0.673908i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38722 + 0.673908i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.822 - 2.88i)T \) |
| 5 | \( 1 + (1.93 - 1.11i)T \) |
| good | 2 | \( 1 + (-1.58 - 0.916i)T + (2 + 3.46i)T^{2} \) |
| 7 | \( 1 + (-3.23 + 5.60i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-9.24 - 5.33i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (5.53 + 9.58i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 12.6iT - 289T^{2} \) |
| 19 | \( 1 + 32.1T + 361T^{2} \) |
| 23 | \( 1 + (9.82 - 5.67i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-35.2 - 20.3i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-15.9 - 27.6i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 45.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (25.4 - 14.7i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-18.8 + 32.5i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (49.8 + 28.7i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 33.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-3.54 + 2.04i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-33.1 + 57.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (35.4 + 61.4i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 13.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 109.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (49.8 - 86.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-74.2 - 42.8i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 63.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (43.2 - 74.8i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−15.31622059440931445210601026271, −14.75482258619514211271326617000, −13.94014491217329445624002806815, −12.50520954405285707955489869418, −10.80216990337738522949520741650, −9.939425591554939362005330657052, −8.279842895399210044471446360698, −6.57109059224895266463400138562, −4.80956227194023513523579579499, −3.85373148466756350820565388570,
2.44313216433727405585032362455, 4.39423308679163680552813884737, 6.24779339727667740798289084924, 8.059508382536309697744419554106, 8.943133951395822967872937777493, 11.53306416550165717953187156171, 11.94646507476768900616886352786, 13.02567048056598265736665270863, 14.12385810589630522062999688936, 14.90107150398709567797252895915
