L(s) = 1 | + (0.819 − 1.82i)2-s − 3.92i·3-s + (−2.65 − 2.98i)4-s + 7.88·5-s + (−7.16 − 3.21i)6-s − 2.24i·7-s + (−7.63 + 2.40i)8-s − 6.42·9-s + (6.45 − 14.3i)10-s + 17.9i·11-s + (−11.7 + 10.4i)12-s + 7.68·13-s + (−4.10 − 1.84i)14-s − 30.9i·15-s + (−1.86 + 15.8i)16-s − 23.5·17-s + ⋯ |
L(s) = 1 | + (0.409 − 0.912i)2-s − 1.30i·3-s + (−0.664 − 0.747i)4-s + 1.57·5-s + (−1.19 − 0.536i)6-s − 0.321i·7-s + (−0.953 + 0.300i)8-s − 0.714·9-s + (0.645 − 1.43i)10-s + 1.63i·11-s + (−0.978 + 0.870i)12-s + 0.590·13-s + (−0.293 − 0.131i)14-s − 2.06i·15-s + (−0.116 + 0.993i)16-s − 1.38·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.683869 - 1.79801i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.683869 - 1.79801i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.819 + 1.82i)T \) |
| 31 | \( 1 - 5.56iT \) |
good | 3 | \( 1 + 3.92iT - 9T^{2} \) |
| 5 | \( 1 - 7.88T + 25T^{2} \) |
| 7 | \( 1 + 2.24iT - 49T^{2} \) |
| 11 | \( 1 - 17.9iT - 121T^{2} \) |
| 13 | \( 1 - 7.68T + 169T^{2} \) |
| 17 | \( 1 + 23.5T + 289T^{2} \) |
| 19 | \( 1 - 8.96iT - 361T^{2} \) |
| 23 | \( 1 + 7.13iT - 529T^{2} \) |
| 29 | \( 1 - 44.0T + 841T^{2} \) |
| 37 | \( 1 + 36.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 4.41T + 1.68e3T^{2} \) |
| 43 | \( 1 + 40.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 7.73iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 85.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 73.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 8.82T + 3.72e3T^{2} \) |
| 67 | \( 1 + 94.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 44.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 118. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 93.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 57.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + 58.0T + 9.40e3T^{2} \) |
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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
−12.77068523902415619613716560485, −12.18954038075790483396696914655, −10.66640237988946878382565148390, −9.870049840000355795602444684625, −8.728445674907538435312887830695, −6.94092793491312197548020615629, −6.10965080089285094626375977402, −4.63905254843006187195551092672, −2.34188182022465032987123751045, −1.51601408818767410948756840322,
3.09562785585486101573575422602, 4.64742514072313790635513454994, 5.71185845246253821852400708956, 6.46946465033360503794154761703, 8.651222966234719650721483181289, 9.102515719691896640962444580219, 10.25356867228709556047917535818, 11.31159198664108403782051273666, 13.10928114351905595491210346098, 13.72916729372691801527455667437