| L(s) = 1 | + 0.850i·2-s + 15.2·4-s + 11.2·5-s − 26.4·7-s + 26.5i·8-s + 9.58i·10-s + 19.9i·11-s + 190. i·13-s − 22.4i·14-s + 221.·16-s + 159.·17-s − 294.·19-s + 172.·20-s − 17.0·22-s − 165. i·23-s + ⋯ |
| L(s) = 1 | + 0.212i·2-s + 0.954·4-s + 0.450·5-s − 0.539·7-s + 0.415i·8-s + 0.0958i·10-s + 0.165i·11-s + 1.12i·13-s − 0.114i·14-s + 0.866·16-s + 0.551·17-s − 0.814·19-s + 0.430·20-s − 0.0351·22-s − 0.312i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.149702935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.149702935\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + (-1.20e3 - 3.26e3i)T \) |
| good | 2 | \( 1 - 0.850iT - 16T^{2} \) |
| 5 | \( 1 - 11.2T + 625T^{2} \) |
| 7 | \( 1 + 26.4T + 2.40e3T^{2} \) |
| 11 | \( 1 - 19.9iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 190. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 159.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 294.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 165. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 513.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.62e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.77e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.06e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.18e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 4.25e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 246.T + 7.89e6T^{2} \) |
| 61 | \( 1 + 1.49e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 3.88e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 1.86e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 619. iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 2.37e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 4.60e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 8.91e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.37e3iT - 8.85e7T^{2} \) |
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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
−10.44982225417584145691552128637, −9.735576618803864668385106856657, −8.735776413664171134192373275658, −7.67606754027548422941422594749, −6.67292410640000625327918590302, −6.22420926487248743687742416241, −5.04304739991793005345988050229, −3.66293201893199267657536204476, −2.45735632835045047912925753610, −1.47373042341553110901926494703,
0.49541987702560929636069595269, 1.93618687756950130535601724151, 2.91110877463652957767226851636, 3.95771434613406292415059870533, 5.73262598717287906219887336964, 6.03528186696741975548373591357, 7.32061456227683046203853278620, 7.980645612069227410608092320746, 9.343714769925615999858461221483, 10.04958436576268373227053460870
