L(s) = 1 | + (0.258 + 0.965i)2-s + (0.965 + 0.258i)3-s + (−0.866 + 0.499i)4-s + i·6-s + (−0.258 − 2.63i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (−2.54 − 4.40i)11-s + (−0.965 + 0.258i)12-s + (−2.02 + 2.02i)13-s + (2.47 − 0.931i)14-s + (0.500 − 0.866i)16-s + (1.79 − 6.71i)17-s + (−0.258 + 0.965i)18-s + (1.79 − 3.11i)19-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (0.557 + 0.149i)3-s + (−0.433 + 0.249i)4-s + 0.408i·6-s + (−0.0978 − 0.995i)7-s + (−0.249 − 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.767 − 1.32i)11-s + (−0.278 + 0.0747i)12-s + (−0.561 + 0.561i)13-s + (0.661 − 0.248i)14-s + (0.125 − 0.216i)16-s + (0.436 − 1.62i)17-s + (−0.0610 + 0.227i)18-s + (0.412 − 0.714i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.708042320\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.708042320\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.258 + 2.63i)T \) |
good | 11 | \( 1 + (2.54 + 4.40i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.02 - 2.02i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.79 + 6.71i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.79 + 3.11i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.71 + 1.79i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 8.81iT - 29T^{2} \) |
| 31 | \( 1 + (-1.61 + 0.931i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.01 + 7.50i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 4.13iT - 41T^{2} \) |
| 43 | \( 1 + (7.84 + 7.84i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.79 + 0.482i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.235 - 0.879i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.08 - 3.61i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.08 + 2.93i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.84 + 0.762i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 7.86T + 71T^{2} \) |
| 73 | \( 1 + (-13.0 - 3.50i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-10.4 - 6.04i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.99 + 1.99i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.82 - 6.62i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (13.4 + 13.4i)T + 97iT^{2} \) |
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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
−9.588426135163003992405501296120, −8.989475604753689231652156284484, −8.111308118056110579056420363712, −7.16456393661783474229553004285, −6.87587249290958964543926988494, −5.31858621647870838391903906123, −4.82875906728705310305853780412, −3.52090103569279188991933569078, −2.81300830903124674697477117471, −0.68956279596837705559118222272,
1.65455521021482851020229349463, 2.56756818529853604154226583848, 3.46512566664849406511948217960, 4.71626179472787527563506887590, 5.48030539014469408793878053058, 6.54994649834328838976271118347, 7.83633169664063859852464102093, 8.232030855343620461155072178840, 9.410527924151321017811238116713, 9.922425882380402458779001271083