| L(s) = 1 | + 5.77i·2-s − 17.3·4-s − 34.3·5-s − 39.2·7-s − 7.56i·8-s − 198. i·10-s − 56.8i·11-s − 232. i·13-s − 226. i·14-s − 233.·16-s + 391.·17-s − 352.·19-s + 594.·20-s + 328.·22-s + 173. i·23-s + ⋯ |
| L(s) = 1 | + 1.44i·2-s − 1.08·4-s − 1.37·5-s − 0.801·7-s − 0.118i·8-s − 1.98i·10-s − 0.469i·11-s − 1.37i·13-s − 1.15i·14-s − 0.911·16-s + 1.35·17-s − 0.976·19-s + 1.48·20-s + 0.677·22-s + 0.327i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8653730655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8653730655\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + (-1.54e3 - 3.11e3i)T \) |
| good | 2 | \( 1 - 5.77iT - 16T^{2} \) |
| 5 | \( 1 + 34.3T + 625T^{2} \) |
| 7 | \( 1 + 39.2T + 2.40e3T^{2} \) |
| 11 | \( 1 + 56.8iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 232. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 391.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 352.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 173. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 90.7T + 7.07e5T^{2} \) |
| 31 | \( 1 - 128. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 413. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 761.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 874. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 895. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.68e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 1.49e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 3.68e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 9.23e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 4.30e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 4.70e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 398. iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.24e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 397. iT - 8.85e7T^{2} \) |
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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
−10.48011811031733148120328794605, −9.341330281463672888115498898550, −8.131721564528317582275096107041, −7.963893997203971791185078124764, −6.98462439807403582953213065565, −6.04245166354146505590804899819, −5.22092225597115980434623048742, −3.94100522035259441883242625511, −3.02797773200840488285618280250, −0.54940282005125941819276914459,
0.44513805285592445477714140704, 1.84198149119784282740384671471, 3.12467165263479809448324491543, 3.89893668712345492369787419523, 4.62691385213680540385224163276, 6.41429509906248093289086099107, 7.29302131644163749341991603139, 8.382592043272924879558232063794, 9.411250839539346327222017869092, 10.04569443848631970577121168567
