| L(s) = 1 | + (−3.83 + 7.96i)2-s + (−16.1 − 12.9i)3-s + (−28.7 − 36.0i)4-s + (32.0 + 15.4i)5-s + (164. − 79.4i)6-s + (44.5 − 55.8i)7-s + (121. − 27.7i)8-s + (41.4 + 181. i)9-s + (−245. + 195. i)10-s + (498. + 113. i)11-s + 955. i·12-s + (178. − 780. i)13-s + (274. + 569. i)14-s + (−319. − 663. i)15-s + (82.7 − 362. i)16-s − 1.16e3i·17-s + ⋯ |
| L(s) = 1 | + (−0.677 + 1.40i)2-s + (−1.03 − 0.828i)3-s + (−0.898 − 1.12i)4-s + (0.572 + 0.275i)5-s + (1.87 − 0.901i)6-s + (0.343 − 0.431i)7-s + (0.672 − 0.153i)8-s + (0.170 + 0.747i)9-s + (−0.776 + 0.619i)10-s + (1.24 + 0.283i)11-s + 1.91i·12-s + (0.292 − 1.28i)13-s + (0.373 + 0.776i)14-s + (−0.366 − 0.761i)15-s + (0.0808 − 0.354i)16-s − 0.981i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0185i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.765567 - 0.00710100i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.765567 - 0.00710100i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (4.40e3 + 1.05e3i)T \) |
| good | 2 | \( 1 + (3.83 - 7.96i)T + (-19.9 - 25.0i)T^{2} \) |
| 3 | \( 1 + (16.1 + 12.9i)T + (54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (-32.0 - 15.4i)T + (1.94e3 + 2.44e3i)T^{2} \) |
| 7 | \( 1 + (-44.5 + 55.8i)T + (-3.73e3 - 1.63e4i)T^{2} \) |
| 11 | \( 1 + (-498. - 113. i)T + (1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-178. + 780. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + 1.16e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + (223. - 177. i)T + (5.50e5 - 2.41e6i)T^{2} \) |
| 23 | \( 1 + (-3.22e3 + 1.55e3i)T + (4.01e6 - 5.03e6i)T^{2} \) |
| 31 | \( 1 + (3.11e3 - 6.46e3i)T + (-1.78e7 - 2.23e7i)T^{2} \) |
| 37 | \( 1 + (-9.81e3 + 2.24e3i)T + (6.24e7 - 3.00e7i)T^{2} \) |
| 41 | \( 1 + 1.55e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (3.49e3 + 7.25e3i)T + (-9.16e7 + 1.14e8i)T^{2} \) |
| 47 | \( 1 + (-2.00e4 - 4.56e3i)T + (2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (1.25e4 + 6.05e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + 1.39e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + (-6.14e3 - 4.90e3i)T + (1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (8.35e3 + 3.66e4i)T + (-1.21e9 + 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-5.68e3 + 2.49e4i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (6.59e3 + 1.36e4i)T + (-1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 + (1.62e4 - 3.70e3i)T + (2.77e9 - 1.33e9i)T^{2} \) |
| 83 | \( 1 + (-3.71e4 - 4.65e4i)T + (-8.76e8 + 3.84e9i)T^{2} \) |
| 89 | \( 1 + (3.60e4 - 7.47e4i)T + (-3.48e9 - 4.36e9i)T^{2} \) |
| 97 | \( 1 + (1.33e5 - 1.06e5i)T + (1.91e9 - 8.37e9i)T^{2} \) |
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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
−16.53433558758623654366879041096, −15.04066249302006490525264731202, −13.91273292870237976274652533153, −12.36434380166878077495818102864, −10.83523681062617623223451594177, −9.190479553692675298248866851213, −7.47731760147851764424190028700, −6.57214846342905759037794029510, −5.46619767265380668242137949060, −0.78061736144608230652212651837,
1.54283681037181489113880465811, 4.10401249230114913696047767254, 5.96793589811218405225544603983, 8.970181962628200786763554100322, 9.711711832135617596803620949293, 11.28180749769930604289588023186, 11.48726209711932972927749193826, 13.08649065017753297890142858121, 14.94660838454837146200782383993, 16.81648611123774918456348634729
