| 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
−16.55324015825257399344353381042, −15.33071480249207822957820992170, −13.45244184922547532418360261618, −12.59703430098818820484172006970, −11.66759255997943886781083581990, −9.932057516086582159247783576849, −8.722106032409934909261256176621, −6.69848664470123982122710712185, −4.48112521299865111397829153982, −2.12308530018701969587805885242,
3.25511530389909712603102810997, 6.31031893171862391675782785655, 6.65321077127289419853436204967, 8.792293957565054899248941110902, 10.53615217913558800844687212980, 11.55304522797465851386390927128, 13.78210730652068544293835121412, 14.40348176757599470490986633141, 15.53706140356639145889968955673, 16.65770113003818346967613184751
