| L(s) = 1 | + (1.45 + 1.28i)2-s + (0.952 + 1.44i)3-s + (0.223 + 1.75i)4-s + (−1.20 − 0.174i)5-s + (−0.468 + 3.33i)6-s + (0.312 + 3.45i)7-s + (0.249 − 0.368i)8-s + (−1.18 + 2.75i)9-s + (−1.52 − 1.79i)10-s + (1.29 − 1.32i)11-s + (−2.32 + 1.99i)12-s + (−1.27 − 0.0461i)13-s + (−3.97 + 5.42i)14-s + (−0.891 − 1.90i)15-s + (4.26 − 1.10i)16-s + (0.435 − 0.201i)17-s + ⋯ |
| L(s) = 1 | + (1.03 + 0.907i)2-s + (0.550 + 0.835i)3-s + (0.111 + 0.878i)4-s + (−0.537 − 0.0781i)5-s + (−0.191 + 1.35i)6-s + (0.118 + 1.30i)7-s + (0.0883 − 0.130i)8-s + (−0.394 + 0.918i)9-s + (−0.482 − 0.568i)10-s + (0.391 − 0.398i)11-s + (−0.672 + 0.576i)12-s + (−0.354 − 0.0128i)13-s + (−1.06 + 1.45i)14-s + (−0.230 − 0.491i)15-s + (1.06 − 0.275i)16-s + (0.105 − 0.0488i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06902 + 2.45827i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06902 + 2.45827i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.952 - 1.44i)T \) |
| 59 | \( 1 + (-7.43 - 1.91i)T \) |
| good | 2 | \( 1 + (-1.45 - 1.28i)T + (0.252 + 1.98i)T^{2} \) |
| 5 | \( 1 + (1.20 + 0.174i)T + (4.79 + 1.42i)T^{2} \) |
| 7 | \( 1 + (-0.312 - 3.45i)T + (-6.88 + 1.25i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 1.32i)T + (-0.198 - 10.9i)T^{2} \) |
| 13 | \( 1 + (1.27 + 0.0461i)T + (12.9 + 0.938i)T^{2} \) |
| 17 | \( 1 + (-0.435 + 0.201i)T + (11.0 - 12.9i)T^{2} \) |
| 19 | \( 1 + (-0.295 - 0.0650i)T + (17.2 + 7.97i)T^{2} \) |
| 23 | \( 1 + (0.705 + 0.266i)T + (17.2 + 15.2i)T^{2} \) |
| 29 | \( 1 + (0.650 + 3.23i)T + (-26.7 + 11.2i)T^{2} \) |
| 31 | \( 1 + (-1.66 + 5.26i)T + (-25.3 - 17.8i)T^{2} \) |
| 37 | \( 1 + (-4.82 - 7.10i)T + (-13.6 + 34.3i)T^{2} \) |
| 41 | \( 1 + (-5.23 - 4.28i)T + (8.08 + 40.1i)T^{2} \) |
| 43 | \( 1 + (1.24 - 0.320i)T + (37.6 - 20.8i)T^{2} \) |
| 47 | \( 1 + (-5.87 + 0.854i)T + (45.0 - 13.3i)T^{2} \) |
| 53 | \( 1 + (-6.79 + 7.99i)T + (-8.57 - 52.3i)T^{2} \) |
| 61 | \( 1 + (-0.556 - 0.490i)T + (7.68 + 60.5i)T^{2} \) |
| 67 | \( 1 + (4.76 + 9.83i)T + (-41.5 + 52.5i)T^{2} \) |
| 71 | \( 1 + (-4.76 + 11.9i)T + (-51.5 - 48.8i)T^{2} \) |
| 73 | \( 1 + (9.86 + 1.07i)T + (71.2 + 15.6i)T^{2} \) |
| 79 | \( 1 + (0.00836 - 0.463i)T + (-78.9 - 2.85i)T^{2} \) |
| 83 | \( 1 + (9.72 - 4.93i)T + (49.0 - 66.9i)T^{2} \) |
| 89 | \( 1 + (-1.51 - 0.510i)T + (70.8 + 53.8i)T^{2} \) |
| 97 | \( 1 + (1.22 + 1.67i)T + (-29.3 + 92.4i)T^{2} \) |
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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
−11.46736210996829569051727588259, −10.10040088426694801354084268633, −9.301333432261692509340046889342, −8.295493926361830838784629579318, −7.66948521472984866629910826856, −6.25604404151238846895340513863, −5.52773407328248117965153340388, −4.59190394814778958431041396956, −3.76803513353678207842992268505, −2.57075383857899260333751988776,
1.22012211126999535598091514247, 2.54895997446380878560322668538, 3.75404202033242178756211713178, 4.25820469290942885357412171586, 5.69853078695906620721002865389, 7.15067880975051586063133314038, 7.51677857068909760182858659610, 8.652769006944807477741510336054, 9.911204114998222202609942308705, 10.82561887632415373008754671654
