| L(s) = 1 | + (3.07 − 2.23i)2-s + (−0.927 − 2.85i)3-s + (1.99 − 6.15i)4-s + (2.08 + 1.51i)5-s + (−9.23 − 6.70i)6-s + (0.454 − 1.39i)7-s + (1.79 + 5.53i)8-s + (−7.28 + 5.29i)9-s + 9.81·10-s + (−7.11 + 35.7i)11-s − 19.4·12-s + (20.4 − 14.8i)13-s + (−1.72 − 5.31i)14-s + (2.39 − 7.35i)15-s + (59.7 + 43.4i)16-s + (−57.4 − 41.7i)17-s + ⋯ |
| L(s) = 1 | + (1.08 − 0.790i)2-s + (−0.178 − 0.549i)3-s + (0.249 − 0.769i)4-s + (0.186 + 0.135i)5-s + (−0.628 − 0.456i)6-s + (0.0245 − 0.0754i)7-s + (0.0794 + 0.244i)8-s + (−0.269 + 0.195i)9-s + 0.310·10-s + (−0.195 + 0.980i)11-s − 0.466·12-s + (0.437 − 0.317i)13-s + (−0.0329 − 0.101i)14-s + (0.0411 − 0.126i)15-s + (0.934 + 0.678i)16-s + (−0.820 − 0.595i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 + 0.917i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.396 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.60657 - 1.05551i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60657 - 1.05551i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.927 + 2.85i)T \) |
| 11 | \( 1 + (7.11 - 35.7i)T \) |
| good | 2 | \( 1 + (-3.07 + 2.23i)T + (2.47 - 7.60i)T^{2} \) |
| 5 | \( 1 + (-2.08 - 1.51i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (-0.454 + 1.39i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-20.4 + 14.8i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (57.4 + 41.7i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-7.55 - 23.2i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-22.2 + 68.4i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-159. + 116. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-73.8 + 227. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-77.5 - 238. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 435.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-153. - 472. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-12.6 + 9.20i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-18.6 + 57.3i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-103. - 75.4i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 889.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (85.7 + 62.3i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-238. + 733. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (898. - 652. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (951. + 691. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 - 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-538. + 391. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−15.74492710878284215356485252526, −14.30743691775020994357420944972, −13.42839606207558139355727118087, −12.40878723106324923247229599616, −11.46047558356022975308913503107, −10.08770002532660780269354156949, −7.935418591705259932597537259248, −6.08888964094089180518060638183, −4.39016487474488471701468751592, −2.30345123669569882701715729987,
3.83985007288244693883992186482, 5.36801250442759266170957059512, 6.51972013457912521764239558955, 8.502267953819553181853713504728, 10.21532945550478841536362601631, 11.72161097201666090698873136424, 13.27368452202389029408959942471, 14.05312726485415006523926060325, 15.35412347499244469698797985726, 16.06088317641472568508235480464
