L(s) = 1 | + (5.96 + 5.96i)3-s + (8.67 − 8.67i)5-s + 1.63i·7-s + 44.1i·9-s + (−18.2 + 18.2i)11-s + (−9.34 − 9.34i)13-s + 103.·15-s + 53.6·17-s + (−70.9 − 70.9i)19-s + (−9.77 + 9.77i)21-s − 25.1i·23-s − 25.6i·25-s + (−102. + 102. i)27-s + (−181. − 181. i)29-s − 132.·31-s + ⋯ |
L(s) = 1 | + (1.14 + 1.14i)3-s + (0.776 − 0.776i)5-s + 0.0885i·7-s + 1.63i·9-s + (−0.498 + 0.498i)11-s + (−0.199 − 0.199i)13-s + 1.78·15-s + 0.764·17-s + (−0.857 − 0.857i)19-s + (−0.101 + 0.101i)21-s − 0.227i·23-s − 0.205i·25-s + (−0.729 + 0.729i)27-s + (−1.15 − 1.15i)29-s − 0.768·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.97913 + 0.734574i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.97913 + 0.734574i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-5.96 - 5.96i)T + 27iT^{2} \) |
| 5 | \( 1 + (-8.67 + 8.67i)T - 125iT^{2} \) |
| 7 | \( 1 - 1.63iT - 343T^{2} \) |
| 11 | \( 1 + (18.2 - 18.2i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (9.34 + 9.34i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 - 53.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + (70.9 + 70.9i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 + 25.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (181. + 181. i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 + 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-174. + 174. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 - 198. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-285. + 285. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + 78.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + (525. - 525. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (46.5 - 46.5i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (-193. - 193. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (282. + 282. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 727. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 106. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 58.9T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-410. - 410. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 768. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 809.T + 9.12e5T^{2} \) |
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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
−14.70125069440133760320680963535, −13.56494627844703598969120736393, −12.66339829950854306001880283379, −10.75702229455889334377378222151, −9.641703767512068370306604675726, −9.052461439264619324268312195253, −7.77714233863238067137603259673, −5.48772838860291919884581925143, −4.22737583341852780965126001110, −2.42938515056317504691844792984,
1.89007037301595103598371103169, 3.22270301478083509587652583664, 5.97074581602943873261856794782, 7.19871704857499675602393597171, 8.218057802697551169751213457741, 9.524231476470720225583141042748, 10.81432943046014674010008291681, 12.48599834639166092794286018010, 13.33296170056693285948989481345, 14.30944608249244471983468256925