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
−16.06088317641472568508235480464, −15.35412347499244469698797985726, −14.05312726485415006523926060325, −13.27368452202389029408959942471, −11.72161097201666090698873136424, −10.21532945550478841536362601631, −8.502267953819553181853713504728, −6.51972013457912521764239558955, −5.36801250442759266170957059512, −3.83985007288244693883992186482,
2.30345123669569882701715729987, 4.39016487474488471701468751592, 6.08888964094089180518060638183, 7.935418591705259932597537259248, 10.08770002532660780269354156949, 11.46047558356022975308913503107, 12.40878723106324923247229599616, 13.42839606207558139355727118087, 14.30743691775020994357420944972, 15.74492710878284215356485252526