L(s) = 1 | + (−0.448 − 0.713i)2-s + (0.187 − 0.534i)3-s + (1.42 − 2.96i)4-s + (4.21 + 0.962i)5-s + (−0.465 + 0.106i)6-s + (−10.1 + 4.88i)7-s + (−6.10 + 0.688i)8-s + (6.78 + 5.41i)9-s + (−1.20 − 3.44i)10-s + (1.54 + 0.173i)11-s + (−1.31 − 1.31i)12-s + (−11.3 + 9.02i)13-s + (8.03 + 5.04i)14-s + (1.30 − 2.07i)15-s + (−4.97 − 6.23i)16-s + (16.8 − 16.8i)17-s + ⋯ |
L(s) = 1 | + (−0.224 − 0.356i)2-s + (0.0623 − 0.178i)3-s + (0.356 − 0.740i)4-s + (0.843 + 0.192i)5-s + (−0.0776 + 0.0177i)6-s + (−1.44 + 0.697i)7-s + (−0.763 + 0.0860i)8-s + (0.753 + 0.601i)9-s + (−0.120 − 0.344i)10-s + (0.140 + 0.0157i)11-s + (−0.109 − 0.109i)12-s + (−0.870 + 0.694i)13-s + (0.574 + 0.360i)14-s + (0.0869 − 0.138i)15-s + (−0.310 − 0.389i)16-s + (0.990 − 0.990i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.895880 - 0.329478i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.895880 - 0.329478i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (20.2 + 20.7i)T \) |
good | 2 | \( 1 + (0.448 + 0.713i)T + (-1.73 + 3.60i)T^{2} \) |
| 3 | \( 1 + (-0.187 + 0.534i)T + (-7.03 - 5.61i)T^{2} \) |
| 5 | \( 1 + (-4.21 - 0.962i)T + (22.5 + 10.8i)T^{2} \) |
| 7 | \( 1 + (10.1 - 4.88i)T + (30.5 - 38.3i)T^{2} \) |
| 11 | \( 1 + (-1.54 - 0.173i)T + (117. + 26.9i)T^{2} \) |
| 13 | \( 1 + (11.3 - 9.02i)T + (37.6 - 164. i)T^{2} \) |
| 17 | \( 1 + (-16.8 + 16.8i)T - 289iT^{2} \) |
| 19 | \( 1 + (-0.448 + 0.157i)T + (282. - 225. i)T^{2} \) |
| 23 | \( 1 + (-1.55 - 6.79i)T + (-476. + 229. i)T^{2} \) |
| 31 | \( 1 + (7.88 + 12.5i)T + (-416. + 865. i)T^{2} \) |
| 37 | \( 1 + (-26.2 + 2.96i)T + (1.33e3 - 304. i)T^{2} \) |
| 41 | \( 1 + (-46.1 - 46.1i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (53.7 + 33.7i)T + (802. + 1.66e3i)T^{2} \) |
| 47 | \( 1 + (-5.72 + 50.7i)T + (-2.15e3 - 491. i)T^{2} \) |
| 53 | \( 1 + (9.09 - 39.8i)T + (-2.53e3 - 1.21e3i)T^{2} \) |
| 59 | \( 1 - 23.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + (-23.8 + 68.2i)T + (-2.90e3 - 2.32e3i)T^{2} \) |
| 67 | \( 1 + (-89.8 - 71.6i)T + (998. + 4.37e3i)T^{2} \) |
| 71 | \( 1 + (54.7 - 43.6i)T + (1.12e3 - 4.91e3i)T^{2} \) |
| 73 | \( 1 + (43.5 - 69.3i)T + (-2.31e3 - 4.80e3i)T^{2} \) |
| 79 | \( 1 + (-12.9 - 114. i)T + (-6.08e3 + 1.38e3i)T^{2} \) |
| 83 | \( 1 + (1.11 + 0.537i)T + (4.29e3 + 5.38e3i)T^{2} \) |
| 89 | \( 1 + (35.1 + 55.9i)T + (-3.43e3 + 7.13e3i)T^{2} \) |
| 97 | \( 1 + (26.0 + 74.3i)T + (-7.35e3 + 5.86e3i)T^{2} \) |
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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
−16.67984733203760040029202454686, −15.63395952112570903291704943856, −14.28150586792387692514690363226, −13.02996580747536615441753619294, −11.71486639997820508220821312607, −9.802342718810230226283374858022, −9.677062492549239675363589616453, −6.95937169255699840165804118049, −5.66115803556563025300955777039, −2.39585652968535222842806348355,
3.47710464042977309130130828067, 6.19925671364488136911232974425, 7.43568178139350007933815925245, 9.349995676823830791360931829897, 10.24243525533813367622202227383, 12.49865590055339195104800168366, 13.02917717468862243703181768199, 14.82821396054989206034015834524, 16.14828737154175583209151110789, 16.90368567272095579466893510064