L(s) = 1 | − 3.23i·3-s + 0.717i·5-s + (0.284 − 2.63i)7-s − 7.45·9-s + (2.98 + 1.43i)11-s − 13-s + 2.31·15-s + 2.69·17-s + 5.82·19-s + (−8.50 − 0.919i)21-s + 8.92·23-s + 4.48·25-s + 14.3i·27-s − 1.83i·29-s − 0.614i·31-s + ⋯ |
L(s) = 1 | − 1.86i·3-s + 0.320i·5-s + (0.107 − 0.994i)7-s − 2.48·9-s + (0.900 + 0.433i)11-s − 0.277·13-s + 0.598·15-s + 0.653·17-s + 1.33·19-s + (−1.85 − 0.200i)21-s + 1.86·23-s + 0.897·25-s + 2.77i·27-s − 0.340i·29-s − 0.110i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.241814737\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.241814737\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.284 + 2.63i)T \) |
| 11 | \( 1 + (-2.98 - 1.43i)T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 3.23iT - 3T^{2} \) |
| 5 | \( 1 - 0.717iT - 5T^{2} \) |
| 17 | \( 1 - 2.69T + 17T^{2} \) |
| 19 | \( 1 - 5.82T + 19T^{2} \) |
| 23 | \( 1 - 8.92T + 23T^{2} \) |
| 29 | \( 1 + 1.83iT - 29T^{2} \) |
| 31 | \( 1 + 0.614iT - 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 9.05T + 41T^{2} \) |
| 43 | \( 1 + 7.96iT - 43T^{2} \) |
| 47 | \( 1 - 3.16iT - 47T^{2} \) |
| 53 | \( 1 - 6.50T + 53T^{2} \) |
| 59 | \( 1 - 4.87iT - 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 0.420T + 67T^{2} \) |
| 71 | \( 1 + 3.11T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 14.7iT - 79T^{2} \) |
| 83 | \( 1 + 9.16T + 83T^{2} \) |
| 89 | \( 1 + 15.4iT - 89T^{2} \) |
| 97 | \( 1 - 3.21iT - 97T^{2} \) |
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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
−7.78437523374961367083973803814, −7.31479456529383265677031333367, −7.04587615205391768024883416738, −6.23895041885364153104460339799, −5.45788611228414050032788904160, −4.41088545010361396329423351373, −3.23027966215639861942018251097, −2.56859527305195416203589705897, −1.23265134449985263565071127005, −0.921781722869329883034932842734,
1.06876960011193142507997815442, 2.95729763558471910023336252121, 3.09448673392311764686738209592, 4.31307990267370927262769882052, 4.87622821039712904105837146710, 5.54395699077592681838997584321, 6.09814661703580747551496452548, 7.31567005656005704201365077977, 8.350743302088698910689217733693, 8.950899837670401253962103564753