| L(s) = 1 | + (−0.881 − 1.10i)2-s + (−0.338 − 1.69i)3-s + (−0.446 + 1.94i)4-s + (0.423 − 0.0370i)5-s + (−1.58 + 1.87i)6-s + (0.916 − 2.51i)7-s + (2.54 − 1.22i)8-s + (−2.77 + 1.15i)9-s + (−0.414 − 0.435i)10-s + (0.274 − 3.14i)11-s + (3.46 + 0.0974i)12-s + (−1.55 + 1.08i)13-s + (−3.59 + 1.20i)14-s + (−0.206 − 0.706i)15-s + (−3.60 − 1.73i)16-s + (1.61 − 2.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.195 − 0.980i)3-s + (−0.223 + 0.974i)4-s + (0.189 − 0.0165i)5-s + (−0.645 + 0.764i)6-s + (0.346 − 0.952i)7-s + (0.901 − 0.433i)8-s + (−0.923 + 0.383i)9-s + (−0.131 − 0.137i)10-s + (0.0828 − 0.946i)11-s + (0.999 + 0.0281i)12-s + (−0.429 + 0.301i)13-s + (−0.960 + 0.322i)14-s + (−0.0532 − 0.182i)15-s + (−0.900 − 0.434i)16-s + (0.392 − 0.679i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0397849 + 0.744303i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0397849 + 0.744303i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.881 + 1.10i)T \) |
| 3 | \( 1 + (0.338 + 1.69i)T \) |
| good | 5 | \( 1 + (-0.423 + 0.0370i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-0.916 + 2.51i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.274 + 3.14i)T + (-10.8 - 1.91i)T^{2} \) |
| 13 | \( 1 + (1.55 - 1.08i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.61 + 2.80i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.408 - 1.52i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (2.45 + 6.75i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.795 + 1.13i)T + (-9.91 - 27.2i)T^{2} \) |
| 31 | \( 1 + (2.03 - 0.740i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.727 - 2.71i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (10.1 + 1.78i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.694 - 7.94i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-6.32 - 2.30i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-0.140 + 0.140i)T - 53iT^{2} \) |
| 59 | \( 1 + (-12.3 + 1.07i)T + (58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (3.35 + 7.18i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-0.433 + 0.303i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (10.2 + 5.89i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.05 + 4.07i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.98 + 16.9i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.39 + 9.13i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-0.920 + 0.531i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.09 + 2.60i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.75839135387936057024896366065, −9.975811184501224803284879016878, −8.777443267548066679144934007755, −7.966060148115010164526212581269, −7.24374826547252614558518823527, −6.17340044478533598034903025849, −4.67529442673905365064602915090, −3.28847914852767441577306894931, −1.94303373723073983939666301840, −0.58977161449520469998539697210,
2.10067554568249007259902749908, 4.01701741600640319199848776918, 5.31479289468769777566608822775, 5.67886889209352672741823041190, 7.02855738082279435809369616036, 8.142395186554258083228175700436, 8.959441467372341948575064357881, 9.800430057919098792578385513956, 10.29206524861628557175460560658, 11.45935988237740689684438895048
