| 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
−11.26762944150660005401374319178, −10.69815291228743834163382654691, −9.665931582358234731617564259977, −8.601634507849822974518051321400, −7.83696782464225796378279339332, −7.03590174634281289666150901208, −5.81074665671256326846647100317, −4.96181156887030610098361569570, −3.88752462472771104559638040682, −1.86271403484116817820175996712,
0.34501787133703191982563881599, 2.02565151852911471097019524725, 3.61261388085889150496395028389, 4.79892303681298725014021072912, 5.77011186723742490389511616231, 7.37118013902087822302684753062, 7.86046133941164999196636759028, 8.681557777021472234504205167804, 9.983470641443984703283108721410, 10.85321378903953407158774823505
