| L(s) = 1 | + (1.66 + 5.13i)2-s + (3.69 − 3.65i)3-s + (−17.1 + 12.4i)4-s + (7.03 + 2.28i)5-s + (24.9 + 12.8i)6-s + (−8.32 − 11.4i)7-s + (−57.6 − 41.8i)8-s + (0.321 − 26.9i)9-s + 39.9i·10-s + (36.4 − 1.15i)11-s + (−17.8 + 108. i)12-s + (−5.34 + 1.73i)13-s + (44.9 − 61.9i)14-s + (34.3 − 17.2i)15-s + (66.6 − 204. i)16-s + (−16.4 + 50.7i)17-s + ⋯ |
| L(s) = 1 | + (0.590 + 1.81i)2-s + (0.711 − 0.702i)3-s + (−2.14 + 1.55i)4-s + (0.628 + 0.204i)5-s + (1.69 + 0.877i)6-s + (−0.449 − 0.618i)7-s + (−2.54 − 1.85i)8-s + (0.0118 − 0.999i)9-s + 1.26i·10-s + (0.999 − 0.0316i)11-s + (−0.429 + 2.61i)12-s + (−0.113 + 0.0370i)13-s + (0.858 − 1.18i)14-s + (0.590 − 0.296i)15-s + (1.04 − 3.20i)16-s + (−0.235 + 0.724i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.12444 + 1.38203i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12444 + 1.38203i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.69 + 3.65i)T \) |
| 11 | \( 1 + (-36.4 + 1.15i)T \) |
| good | 2 | \( 1 + (-1.66 - 5.13i)T + (-6.47 + 4.70i)T^{2} \) |
| 5 | \( 1 + (-7.03 - 2.28i)T + (101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (8.32 + 11.4i)T + (-105. + 326. i)T^{2} \) |
| 13 | \( 1 + (5.34 - 1.73i)T + (1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (16.4 - 50.7i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (20.6 - 28.4i)T + (-2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + 64.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (113. - 82.7i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (9.10 + 28.0i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (171. - 124. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-213. - 154. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 385. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-133. + 184. i)T + (-3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-185. + 60.3i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-277. - 381. i)T + (-6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (616. + 200. i)T + (1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 - 823.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-605. - 196. i)T + (2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-110. - 152. i)T + (-1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-816. + 265. i)T + (3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (67.7 - 208. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 - 217. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (537. + 1.65e3i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−16.57024909561364492314174412168, −15.05410295983619272201417424932, −14.27260899204612011638229312994, −13.47267903510600512979996958119, −12.50845247946089339191128578740, −9.548273582949129392820049553675, −8.315872164751948205628343421971, −6.95587346359383610223798678348, −6.15373412318823339913304769624, −3.88267505425479685082547511715,
2.23810563165073851793245748260, 3.81035632643422602432186820753, 5.40452979343555964731143787694, 9.141093259920038713722568293061, 9.504168683679017238608741868994, 10.90647065404528823875491738208, 12.17411455330946372249326690272, 13.42200763586739863484112781088, 14.18830778030050442621831822215, 15.43751938505640636583124460317
