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
−12.83491430014718585671156536409, −12.12057151165188782550826648721, −11.41722892828938619077520190901, −10.01617620429081015266280104083, −9.137317096982558428508631519488, −8.221135912219623246630480468533, −6.35114173461415016276057747944, −4.87214463881402183088774002466, −3.28983459568688344859573850367, −1.75961764882611012292001948525,
0.78317034622281289097070280781, 3.90722803729044424184236502567, 5.02229856469833426003730812215, 6.82602137479008418459105861417, 7.23703829806877555794924245999, 8.814815546069073166802732382070, 9.688209856611278888574083000385, 10.87908821912150129780229071487, 12.31897117083439183877109997066, 13.58933647804810167792690105445