L(s) = 1 | + (−0.419 + 2.79i)2-s + (−7.64 − 2.34i)4-s − 1.49i·5-s − 26.1i·7-s + (9.78 − 20.4i)8-s + (4.18 + 0.627i)10-s + 56.3·11-s − 41.3·13-s + (73.2 + 10.9i)14-s + (52.9 + 35.9i)16-s − 51.0i·17-s − 79.0i·19-s + (−3.51 + 11.4i)20-s + (−23.6 + 157. i)22-s + 27.3·23-s + ⋯ |
L(s) = 1 | + (−0.148 + 0.988i)2-s + (−0.955 − 0.293i)4-s − 0.133i·5-s − 1.41i·7-s + (0.432 − 0.901i)8-s + (0.132 + 0.0198i)10-s + 1.54·11-s − 0.881·13-s + (1.39 + 0.209i)14-s + (0.827 + 0.561i)16-s − 0.728i·17-s − 0.954i·19-s + (−0.0392 + 0.127i)20-s + (−0.229 + 1.52i)22-s + 0.248·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.21623 - 0.182582i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21623 - 0.182582i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.419 - 2.79i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.49iT - 125T^{2} \) |
| 7 | \( 1 + 26.1iT - 343T^{2} \) |
| 11 | \( 1 - 56.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 51.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 79.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 27.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 134. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 187. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 298. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 465. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 373.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 620. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 321.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 576. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 223.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 70.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.34e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 576.T + 9.12e5T^{2} \) |
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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
−13.58933647804810167792690105445, −12.31897117083439183877109997066, −10.87908821912150129780229071487, −9.688209856611278888574083000385, −8.814815546069073166802732382070, −7.23703829806877555794924245999, −6.82602137479008418459105861417, −5.02229856469833426003730812215, −3.90722803729044424184236502567, −0.78317034622281289097070280781,
1.75961764882611012292001948525, 3.28983459568688344859573850367, 4.87214463881402183088774002466, 6.35114173461415016276057747944, 8.221135912219623246630480468533, 9.137317096982558428508631519488, 10.01617620429081015266280104083, 11.41722892828938619077520190901, 12.12057151165188782550826648721, 12.83491430014718585671156536409