| L(s) = 1 | + (−0.669 − 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (−1.47 − 0.313i)5-s + (0.309 + 0.951i)6-s + (2.56 − 0.633i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (0.753 + 1.30i)10-s + (−3.27 − 0.552i)11-s + (0.5 − 0.866i)12-s + (1.02 − 3.16i)13-s + (−2.18 − 1.48i)14-s + (1.21 + 0.886i)15-s + (−0.978 − 0.207i)16-s + (−0.677 + 0.752i)17-s + ⋯ |
| L(s) = 1 | + (−0.473 − 0.525i)2-s + (−0.527 − 0.234i)3-s + (−0.0522 + 0.497i)4-s + (−0.659 − 0.140i)5-s + (0.126 + 0.388i)6-s + (0.970 − 0.239i)7-s + (0.286 − 0.207i)8-s + (0.223 + 0.247i)9-s + (0.238 + 0.412i)10-s + (−0.986 − 0.166i)11-s + (0.144 − 0.249i)12-s + (0.285 − 0.878i)13-s + (−0.585 − 0.396i)14-s + (0.314 + 0.228i)15-s + (−0.244 − 0.0519i)16-s + (−0.164 + 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0418696 - 0.454964i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0418696 - 0.454964i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.669 + 0.743i)T \) |
| 3 | \( 1 + (0.913 + 0.406i)T \) |
| 7 | \( 1 + (-2.56 + 0.633i)T \) |
| 11 | \( 1 + (3.27 + 0.552i)T \) |
| good | 5 | \( 1 + (1.47 + 0.313i)T + (4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (-1.02 + 3.16i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.677 - 0.752i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.447 + 4.25i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (2.00 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (7.79 + 5.66i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (3.05 - 0.650i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (2.91 - 1.29i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (-3.91 + 2.84i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 8.86T + 43T^{2} \) |
| 47 | \( 1 + (0.256 + 2.44i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-0.785 + 0.167i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.439 - 4.17i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (-5.10 - 1.08i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (0.962 + 1.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.87 + 11.9i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.558 + 5.31i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-9.47 - 10.5i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (3.24 + 9.97i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-2.34 + 4.05i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.904 + 2.78i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.85321378903953407158774823505, −9.983470641443984703283108721410, −8.681557777021472234504205167804, −7.86046133941164999196636759028, −7.37118013902087822302684753062, −5.77011186723742490389511616231, −4.79892303681298725014021072912, −3.61261388085889150496395028389, −2.02565151852911471097019524725, −0.34501787133703191982563881599,
1.86271403484116817820175996712, 3.88752462472771104559638040682, 4.96181156887030610098361569570, 5.81074665671256326846647100317, 7.03590174634281289666150901208, 7.83696782464225796378279339332, 8.601634507849822974518051321400, 9.665931582358234731617564259977, 10.69815291228743834163382654691, 11.26762944150660005401374319178
