| 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
−10.23626322199606304996389118702, −9.347104670212587798372847022138, −8.711855673029945783562766226279, −7.42270317584503985797844534163, −6.32274674506805537275007045246, −5.38501659860608161006795488747, −4.42037526679983200808677559961, −3.95469336381601094555952319338, −2.70018610432835746347267991117, −0.02987927517199478563245586930,
2.34929253256640943484792425340, 2.94278203153652236225367420258, 4.55509505577737151784412890822, 5.35265222089853431172202406068, 6.29319299755342567929590126807, 7.23494369508148462799109637143, 8.246150385290020069714386492260, 9.013365093308381125418082796203, 9.783727627697611879378426744204, 11.27655232274696771791001046737
