| L(s) = 1 | + (−2.12 − 1.48i)2-s + (1.59 − 0.685i)3-s + (1.60 + 4.41i)4-s + (−2.09 − 0.783i)5-s + (−4.39 − 0.908i)6-s + (−1.25 − 2.68i)7-s + (1.80 − 6.75i)8-s + (2.06 − 2.18i)9-s + (3.27 + 4.77i)10-s + (−1.00 − 1.19i)11-s + (5.58 + 5.92i)12-s + (−2.29 − 3.28i)13-s + (−1.33 + 7.54i)14-s + (−3.86 + 0.188i)15-s + (−6.66 + 5.59i)16-s + (0.490 + 1.82i)17-s + ⋯ |
| L(s) = 1 | + (−1.49 − 1.04i)2-s + (0.918 − 0.395i)3-s + (0.803 + 2.20i)4-s + (−0.936 − 0.350i)5-s + (−1.79 − 0.370i)6-s + (−0.473 − 1.01i)7-s + (0.639 − 2.38i)8-s + (0.686 − 0.726i)9-s + (1.03 + 1.50i)10-s + (−0.302 − 0.360i)11-s + (1.61 + 1.71i)12-s + (−0.637 − 0.910i)13-s + (−0.355 + 2.01i)14-s + (−0.998 + 0.0486i)15-s + (−1.66 + 1.39i)16-s + (0.118 + 0.443i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.113651 - 0.524334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.113651 - 0.524334i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.59 + 0.685i)T \) |
| 5 | \( 1 + (2.09 + 0.783i)T \) |
| good | 2 | \( 1 + (2.12 + 1.48i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (1.25 + 2.68i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (1.00 + 1.19i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.29 + 3.28i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.490 - 1.82i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (2.41 - 1.39i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.78 - 3.16i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.683 + 3.87i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-5.45 + 1.98i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (0.316 - 0.0847i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-6.26 - 1.10i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.0509 + 0.582i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (0.690 - 0.321i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (5.57 + 5.57i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.84 - 6.57i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-11.4 - 4.15i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.63 - 1.14i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (6.11 + 3.52i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.93 - 1.05i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (12.9 - 2.28i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.33 - 3.33i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-4.23 - 7.33i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.95 - 0.696i)T + (95.5 + 16.8i)T^{2} \) |
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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
−12.73303123058110807059446083631, −11.59630328127524661754843786378, −10.50968201328404381064925838457, −9.705517140811174508264971972467, −8.556419161369038035709840385702, −7.84473960806759679598091820123, −7.11031008647870213833086232189, −3.92074934935960596831002497732, −2.87700359378077299091166906912, −0.822459878182114521103236497910,
2.61412424929225251312703808893, 4.84257526240806640029766621684, 6.68664018737742380519105080976, 7.44275637362406827702454174373, 8.576736106541705977688008770234, 9.131683550474385818481316852945, 10.10707715659021624528734932683, 11.19729055839157083742492925352, 12.62677817240673420790031127099, 14.42890237075512507935078406097
