| L(s) = 1 | + (0.637 − 0.307i)2-s + (−1.26 + 1.18i)3-s + (−0.934 + 1.17i)4-s + (−1.99 + 0.614i)5-s + (−0.439 + 1.14i)6-s + (2.89 + 1.67i)7-s + (−0.551 + 2.41i)8-s + (0.181 − 2.99i)9-s + (−1.08 + 1.00i)10-s + (−2.61 + 2.08i)11-s + (−0.212 − 2.58i)12-s + (2.41 + 2.23i)13-s + (2.36 + 0.177i)14-s + (1.78 − 3.14i)15-s + (−0.277 − 1.21i)16-s + (1.50 − 4.88i)17-s + ⋯ |
| L(s) = 1 | + (0.450 − 0.217i)2-s + (−0.728 + 0.685i)3-s + (−0.467 + 0.586i)4-s + (−0.891 + 0.274i)5-s + (−0.179 + 0.467i)6-s + (1.09 + 0.632i)7-s + (−0.194 + 0.853i)8-s + (0.0606 − 0.998i)9-s + (−0.342 + 0.317i)10-s + (−0.787 + 0.628i)11-s + (−0.0612 − 0.747i)12-s + (0.669 + 0.621i)13-s + (0.631 + 0.0473i)14-s + (0.460 − 0.810i)15-s + (−0.0693 − 0.303i)16-s + (0.365 − 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0840 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0840 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.585646 + 0.637121i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.585646 + 0.637121i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.26 - 1.18i)T \) |
| 43 | \( 1 + (4.62 + 4.64i)T \) |
| good | 2 | \( 1 + (-0.637 + 0.307i)T + (1.24 - 1.56i)T^{2} \) |
| 5 | \( 1 + (1.99 - 0.614i)T + (4.13 - 2.81i)T^{2} \) |
| 7 | \( 1 + (-2.89 - 1.67i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.61 - 2.08i)T + (2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.41 - 2.23i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.50 + 4.88i)T + (-14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-0.553 + 3.67i)T + (-18.1 - 5.60i)T^{2} \) |
| 23 | \( 1 + (-5.58 - 2.19i)T + (16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (0.164 - 2.19i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (-6.48 - 4.42i)T + (11.3 + 28.8i)T^{2} \) |
| 37 | \( 1 + (1.81 - 1.04i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.79 - 5.80i)T + (-25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (-5.46 - 4.35i)T + (10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (8.49 + 9.15i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (-5.38 + 1.23i)T + (53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (-0.781 - 1.14i)T + (-22.2 + 56.7i)T^{2} \) |
| 67 | \( 1 + (2.47 + 0.373i)T + (64.0 + 19.7i)T^{2} \) |
| 71 | \( 1 + (0.681 + 1.73i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (1.97 - 2.12i)T + (-5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (-5.69 + 9.87i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-16.4 + 1.23i)T + (82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (-0.426 - 5.69i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (12.2 + 15.3i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−13.52614051068119478706264496926, −12.23620739954840572293160913875, −11.57242743295156763598251235171, −11.02515120094233695800780438278, −9.391504517583944984272125231008, −8.343379015868382982172874211472, −7.14793211011261089072259261604, −5.15871449070410105492397723010, −4.63768270137012858534775545834, −3.15885965160942782930666589841,
0.971272332303623404171352167829, 4.05620856361066034792306503984, 5.19265925775507284398739592145, 6.16979161262305892894744682188, 7.76687924554910215703977029332, 8.323545868768378840185319352231, 10.43571422449083260573077663695, 10.97418284523249978146061444971, 12.18271559355076645061782262620, 13.13730357543231023958235119597
