L(s) = 1 | + (2.97 − 2.67i)2-s + (1.73 − 15.9i)4-s + (2.84 + 2.84i)5-s − 76.7·7-s + (−37.2 − 52.0i)8-s + (16.0 + 0.876i)10-s + (−121. + 121. i)11-s + (27.1 − 27.1i)13-s + (−228. + 205. i)14-s + (−249. − 55.2i)16-s + 88.0·17-s + (−261. − 261. i)19-s + (50.2 − 40.3i)20-s + (−37.3 + 686. i)22-s − 93.4·23-s + ⋯ |
L(s) = 1 | + (0.744 − 0.667i)2-s + (0.108 − 0.994i)4-s + (0.113 + 0.113i)5-s − 1.56·7-s + (−0.582 − 0.812i)8-s + (0.160 + 0.00876i)10-s + (−1.00 + 1.00i)11-s + (0.160 − 0.160i)13-s + (−1.16 + 1.04i)14-s + (−0.976 − 0.215i)16-s + 0.304·17-s + (−0.723 − 0.723i)19-s + (0.125 − 0.100i)20-s + (−0.0772 + 1.41i)22-s − 0.176·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.145005 + 0.460270i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.145005 + 0.460270i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.97 + 2.67i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.84 - 2.84i)T + 625iT^{2} \) |
| 7 | \( 1 + 76.7T + 2.40e3T^{2} \) |
| 11 | \( 1 + (121. - 121. i)T - 1.46e4iT^{2} \) |
| 13 | \( 1 + (-27.1 + 27.1i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 - 88.0T + 8.35e4T^{2} \) |
| 19 | \( 1 + (261. + 261. i)T + 1.30e5iT^{2} \) |
| 23 | \( 1 + 93.4T + 2.79e5T^{2} \) |
| 29 | \( 1 + (272. - 272. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 - 1.23e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (1.04e3 + 1.04e3i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 - 915. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-1.11e3 + 1.11e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + 1.72e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (734. + 734. i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 + (-1.20e3 + 1.20e3i)T - 1.21e7iT^{2} \) |
| 61 | \( 1 + (-580. + 580. i)T - 1.38e7iT^{2} \) |
| 67 | \( 1 + (1.48e3 + 1.48e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 + 5.57e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 6.61e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 5.39e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-2.55e3 - 2.55e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + 1.09e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 4.71e3T + 8.85e7T^{2} \) |
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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
−12.15818207622230604125471008666, −10.58498878183551913886042711297, −10.12012442187768343398443195878, −9.018429410326602439883325111636, −7.13086295684187932913852567654, −6.16560015959152423894303746982, −4.88850361491513857296205031500, −3.44954593440434304466378904757, −2.31560487697406706104674077743, −0.13712263965820740763824016922,
2.82227905672313522555408653171, 3.87902660999495751349244115950, 5.59859777713100856511214848741, 6.26683874012451269636621573973, 7.54093505206554800563375253844, 8.663411743176057724102866859506, 9.854745456147501762962729782564, 11.13073643993942495175486234591, 12.42804803281566492831575059824, 13.14950613001385045978985396561