L(s) = 1 | + (20.0 + 20.0i)2-s + (6.21 + 6.21i)3-s + 549. i·4-s + 611. i·5-s + 249. i·6-s + 794.·7-s + (−5.90e3 + 5.90e3i)8-s − 6.48e3i·9-s + (−1.22e4 + 1.22e4i)10-s + (−5.32e3 − 5.32e3i)11-s + (−3.41e3 + 3.41e3i)12-s + 4.41e4i·13-s + (1.59e4 + 1.59e4i)14-s + (−3.80e3 + 3.80e3i)15-s − 9.61e4·16-s + (1.57e4 + 1.57e4i)17-s + ⋯ |
L(s) = 1 | + (1.25 + 1.25i)2-s + (0.0767 + 0.0767i)3-s + 2.14i·4-s + 0.979i·5-s + 0.192i·6-s + 0.330·7-s + (−1.44 + 1.44i)8-s − 0.988i·9-s + (−1.22 + 1.22i)10-s + (−0.363 − 0.363i)11-s + (−0.164 + 0.164i)12-s + 1.54i·13-s + (0.415 + 0.415i)14-s + (−0.0751 + 0.0751i)15-s − 1.46·16-s + (0.188 + 0.188i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.684862 + 3.30045i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.684862 + 3.30045i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (-5.83e5 - 3.99e5i)T \) |
good | 2 | \( 1 + (-20.0 - 20.0i)T + 256iT^{2} \) |
| 3 | \( 1 + (-6.21 - 6.21i)T + 6.56e3iT^{2} \) |
| 5 | \( 1 - 611. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 794.T + 5.76e6T^{2} \) |
| 11 | \( 1 + (5.32e3 + 5.32e3i)T + 2.14e8iT^{2} \) |
| 13 | \( 1 - 4.41e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-1.57e4 - 1.57e4i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 + (1.00e5 + 1.00e5i)T + 1.69e10iT^{2} \) |
| 23 | \( 1 - 3.94e5T + 7.83e10T^{2} \) |
| 31 | \( 1 + (-2.05e5 - 2.05e5i)T + 8.52e11iT^{2} \) |
| 37 | \( 1 + (-8.02e4 + 8.02e4i)T - 3.51e12iT^{2} \) |
| 41 | \( 1 + (-1.98e6 + 1.98e6i)T - 7.98e12iT^{2} \) |
| 43 | \( 1 + (-8.46e5 - 8.46e5i)T + 1.16e13iT^{2} \) |
| 47 | \( 1 + (-1.49e6 + 1.49e6i)T - 2.38e13iT^{2} \) |
| 53 | \( 1 + 1.06e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.90e7T + 1.46e14T^{2} \) |
| 61 | \( 1 + (-1.26e7 - 1.26e7i)T + 1.91e14iT^{2} \) |
| 67 | \( 1 + 2.92e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 2.31e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (-2.90e7 + 2.90e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 + (3.05e7 + 3.05e7i)T + 1.51e15iT^{2} \) |
| 83 | \( 1 + 6.94e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (-7.31e6 - 7.31e6i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 + (5.69e7 - 5.69e7i)T - 7.83e15iT^{2} \) |
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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
−15.40154370524599254693046346212, −14.64961962389028990640815734714, −13.85548036775274921299986817077, −12.46107535736687999023814308491, −11.08826036249979476501379769281, −8.846539009932337089411508870220, −7.09615169558479517792204056589, −6.34514954912879634361765898539, −4.58063083190662679434717131723, −3.11241901336011804870583173329,
1.10247438405348840590544211550, 2.68396633427433844783478265558, 4.58617333301306701584238971566, 5.46279003127761933031905608768, 8.149212171023851642865319173644, 10.10511387834611991447755470708, 11.12605464724031831195682529083, 12.66626884519737591304182215836, 13.00704321306723086113373202039, 14.34686622403756335564672718269