| L(s) = 1 | + (1.03 − 0.183i)2-s + (0.0916 − 1.72i)3-s + (−0.834 + 0.303i)4-s + (−0.221 − 1.81i)6-s + (−0.841 + 2.31i)7-s + (−2.63 + 1.52i)8-s + (−2.98 − 0.317i)9-s + (−0.960 + 0.806i)11-s + (0.449 + 1.47i)12-s + (−4.47 − 0.789i)13-s + (−0.450 + 2.55i)14-s + (−1.09 + 0.921i)16-s + (−5.75 − 3.32i)17-s + (−3.15 + 0.216i)18-s + (0.124 + 0.215i)19-s + ⋯ |
| L(s) = 1 | + (0.734 − 0.129i)2-s + (0.0529 − 0.998i)3-s + (−0.417 + 0.151i)4-s + (−0.0904 − 0.740i)6-s + (−0.317 + 0.873i)7-s + (−0.932 + 0.538i)8-s + (−0.994 − 0.105i)9-s + (−0.289 + 0.243i)11-s + (0.129 + 0.424i)12-s + (−1.24 − 0.219i)13-s + (−0.120 + 0.682i)14-s + (−0.274 + 0.230i)16-s + (−1.39 − 0.806i)17-s + (−0.743 + 0.0511i)18-s + (0.0285 + 0.0495i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0259596 + 0.0728969i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0259596 + 0.0728969i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.0916 + 1.72i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (-1.03 + 0.183i)T + (1.87 - 0.684i)T^{2} \) |
| 7 | \( 1 + (0.841 - 2.31i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.960 - 0.806i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (4.47 + 0.789i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (5.75 + 3.32i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.124 - 0.215i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.287 + 0.791i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.0889 - 0.504i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.770 + 0.280i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.25 - 1.30i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.41 - 8.02i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.78 - 3.31i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.81 + 4.98i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 10.4iT - 53T^{2} \) |
| 59 | \( 1 + (-2.30 - 1.93i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (2.70 + 0.986i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (9.93 + 1.75i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (0.0447 - 0.0774i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.60 - 2.66i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.829 + 4.70i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (7.91 - 1.39i)T + (77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.35 - 5.80i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.52 + 4.20i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.27655232274696771791001046737, −9.783727627697611879378426744204, −9.013365093308381125418082796203, −8.246150385290020069714386492260, −7.23494369508148462799109637143, −6.29319299755342567929590126807, −5.35265222089853431172202406068, −4.55509505577737151784412890822, −2.94278203153652236225367420258, −2.34929253256640943484792425340,
0.02987927517199478563245586930, 2.70018610432835746347267991117, 3.95469336381601094555952319338, 4.42037526679983200808677559961, 5.38501659860608161006795488747, 6.32274674506805537275007045246, 7.42270317584503985797844534163, 8.711855673029945783562766226279, 9.347104670212587798372847022138, 10.23626322199606304996389118702
