| L(s) = 1 | + (0.484 + 2.12i)2-s + (0.796 + 0.383i)3-s + (2.93 − 1.41i)4-s + (3.15 + 13.8i)5-s + (−0.428 + 1.87i)6-s + (−15.6 − 7.53i)7-s + (15.2 + 19.1i)8-s + (−16.3 − 20.4i)9-s + (−27.8 + 13.3i)10-s + (45.2 − 56.7i)11-s + 2.87·12-s + (0.621 − 0.778i)13-s + (8.41 − 36.8i)14-s + (−2.78 + 12.2i)15-s + (−17.0 + 21.4i)16-s − 98.1·17-s + ⋯ |
| L(s) = 1 | + (0.171 + 0.750i)2-s + (0.153 + 0.0738i)3-s + (0.366 − 0.176i)4-s + (0.282 + 1.23i)5-s + (−0.0291 + 0.127i)6-s + (−0.844 − 0.406i)7-s + (0.675 + 0.847i)8-s + (−0.605 − 0.759i)9-s + (−0.879 + 0.423i)10-s + (1.24 − 1.55i)11-s + 0.0691·12-s + (0.0132 − 0.0166i)13-s + (0.160 − 0.704i)14-s + (−0.0480 + 0.210i)15-s + (−0.266 + 0.334i)16-s − 1.40·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 - 0.876i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.482 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.25412 + 0.741195i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25412 + 0.741195i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-150. - 42.4i)T \) |
| good | 2 | \( 1 + (-0.484 - 2.12i)T + (-7.20 + 3.47i)T^{2} \) |
| 3 | \( 1 + (-0.796 - 0.383i)T + (16.8 + 21.1i)T^{2} \) |
| 5 | \( 1 + (-3.15 - 13.8i)T + (-112. + 54.2i)T^{2} \) |
| 7 | \( 1 + (15.6 + 7.53i)T + (213. + 268. i)T^{2} \) |
| 11 | \( 1 + (-45.2 + 56.7i)T + (-296. - 1.29e3i)T^{2} \) |
| 13 | \( 1 + (-0.621 + 0.778i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + 98.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + (28.4 - 13.7i)T + (4.27e3 - 5.36e3i)T^{2} \) |
| 23 | \( 1 + (10.2 - 44.7i)T + (-1.09e4 - 5.27e3i)T^{2} \) |
| 31 | \( 1 + (-46.6 - 204. i)T + (-2.68e4 + 1.29e4i)T^{2} \) |
| 37 | \( 1 + (86.0 + 107. i)T + (-1.12e4 + 4.93e4i)T^{2} \) |
| 41 | \( 1 + 284.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (43.5 - 190. i)T + (-7.16e4 - 3.44e4i)T^{2} \) |
| 47 | \( 1 + (33.4 - 41.8i)T + (-2.31e4 - 1.01e5i)T^{2} \) |
| 53 | \( 1 + (-74.4 - 326. i)T + (-1.34e5 + 6.45e4i)T^{2} \) |
| 59 | \( 1 - 654.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (564. + 271. i)T + (1.41e5 + 1.77e5i)T^{2} \) |
| 67 | \( 1 + (208. + 261. i)T + (-6.69e4 + 2.93e5i)T^{2} \) |
| 71 | \( 1 + (-403. + 506. i)T + (-7.96e4 - 3.48e5i)T^{2} \) |
| 73 | \( 1 + (81.4 - 356. i)T + (-3.50e5 - 1.68e5i)T^{2} \) |
| 79 | \( 1 + (392. + 492. i)T + (-1.09e5 + 4.80e5i)T^{2} \) |
| 83 | \( 1 + (-616. + 296. i)T + (3.56e5 - 4.47e5i)T^{2} \) |
| 89 | \( 1 + (-18.6 - 81.8i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + (14.8 - 7.12i)T + (5.69e5 - 7.13e5i)T^{2} \) |
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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
−16.65770113003818346967613184751, −15.53706140356639145889968955673, −14.40348176757599470490986633141, −13.78210730652068544293835121412, −11.55304522797465851386390927128, −10.53615217913558800844687212980, −8.792293957565054899248941110902, −6.65321077127289419853436204967, −6.31031893171862391675782785655, −3.25511530389909712603102810997,
2.12308530018701969587805885242, 4.48112521299865111397829153982, 6.69848664470123982122710712185, 8.722106032409934909261256176621, 9.932057516086582159247783576849, 11.66759255997943886781083581990, 12.59703430098818820484172006970, 13.45244184922547532418360261618, 15.33071480249207822957820992170, 16.55324015825257399344353381042
