| L(s) = 1 | − 0.389i·2-s + 15.8·4-s + 17.9·5-s − 47.2·7-s − 12.3i·8-s − 7.00i·10-s + 197. i·11-s − 176. i·13-s + 18.3i·14-s + 248.·16-s − 486.·17-s + 56.2·19-s + 285.·20-s + 76.7·22-s + 848. i·23-s + ⋯ |
| L(s) = 1 | − 0.0972i·2-s + 0.990·4-s + 0.719·5-s − 0.964·7-s − 0.193i·8-s − 0.0700i·10-s + 1.63i·11-s − 1.04i·13-s + 0.0938i·14-s + 0.971·16-s − 1.68·17-s + 0.155·19-s + 0.713·20-s + 0.158·22-s + 1.60i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.882i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.471 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.456473453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.456473453\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + (-1.64e3 - 3.07e3i)T \) |
| good | 2 | \( 1 + 0.389iT - 16T^{2} \) |
| 5 | \( 1 - 17.9T + 625T^{2} \) |
| 7 | \( 1 + 47.2T + 2.40e3T^{2} \) |
| 11 | \( 1 - 197. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 176. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 486.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 56.2T + 1.30e5T^{2} \) |
| 23 | \( 1 - 848. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 275.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 843. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.85e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.47e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.75e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.43e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.11e3T + 7.89e6T^{2} \) |
| 61 | \( 1 + 115. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 49.5iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.08e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 209. iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 6.54e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 6.80e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 9.11e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 2.94e3iT - 8.85e7T^{2} \) |
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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
−10.21215229407629391504779378509, −9.991692958825703008624428119314, −8.951858902790220728436021084125, −7.54087152386126736788811554261, −6.88786652510547699996311922224, −6.08599625761252245474140111943, −5.06154838188947864884830301011, −3.56182274774005834512339702730, −2.46731540205499090743325363514, −1.58815905476157319142309553906,
0.31452494382940946077546868910, 1.97095849656835540040022291993, 2.82472630897100231838087601414, 4.05110880800414749389197150689, 5.65441287919658241122548102246, 6.41929539495608904311325539531, 6.79837195319661881673868920582, 8.251699805981362686955564975829, 9.092632170704081539276080774087, 9.954323933722637670741021101323
