L(s) = 1 | − 2.82i·2-s − 8.00·4-s + 33.0·5-s + 16.0·7-s + 22.6i·8-s − 93.6i·10-s + 215.·11-s + 281. i·13-s − 45.4i·14-s + 64.0·16-s − 226.·17-s + (13.9 + 360. i)19-s − 264.·20-s − 609. i·22-s − 414.·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.32·5-s + 0.328·7-s + 0.353i·8-s − 0.936i·10-s + 1.78·11-s + 1.66i·13-s − 0.232i·14-s + 0.250·16-s − 0.784·17-s + (0.0387 + 0.999i)19-s − 0.661·20-s − 1.25i·22-s − 0.783·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.634116696\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.634116696\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-13.9 - 360. i)T \) |
good | 5 | \( 1 - 33.0T + 625T^{2} \) |
| 7 | \( 1 - 16.0T + 2.40e3T^{2} \) |
| 11 | \( 1 - 215.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 281. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 226.T + 8.35e4T^{2} \) |
| 23 | \( 1 + 414.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 606. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 478. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 104. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.89e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 315.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 474.T + 4.87e6T^{2} \) |
| 53 | \( 1 + 774. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 567. iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.79e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 4.46e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 7.63e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.39e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 9.70e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.05e4T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.02e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.54e4iT - 8.85e7T^{2} \) |
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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
−10.99498123448542838416326124940, −9.764996367692060174855181709642, −9.357831800649847512364749993668, −8.484099833354779246454530665011, −6.76124649549760298262508716104, −6.11639215585286157219451552954, −4.71945400422288562323356531063, −3.73901190451142807992977957460, −1.97945500767073243381566866565, −1.48072957676326213467968520091,
0.824933295848586916747602437065, 2.26761502979369915566635992157, 3.93241709228452868257124826810, 5.21924249798903306018100126574, 6.07365400371838037815075462976, 6.82099935431870540443943924846, 8.083313873388965600087298386282, 9.082125832123365227600401686316, 9.711848155191575437938782454491, 10.71492903747871963493916097560