| L(s) = 1 | + (0.669 + 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (−1.55 − 0.330i)5-s + (−0.309 − 0.951i)6-s + (−1.63 − 2.07i)7-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.794 − 1.37i)10-s + (3.30 + 0.278i)11-s + (0.5 − 0.866i)12-s + (1.65 − 5.09i)13-s + (0.451 − 2.60i)14-s + (1.28 + 0.933i)15-s + (−0.978 − 0.207i)16-s + (4.88 − 5.42i)17-s + ⋯ |
| L(s) = 1 | + (0.473 + 0.525i)2-s + (−0.527 − 0.234i)3-s + (−0.0522 + 0.497i)4-s + (−0.695 − 0.147i)5-s + (−0.126 − 0.388i)6-s + (−0.618 − 0.786i)7-s + (−0.286 + 0.207i)8-s + (0.223 + 0.247i)9-s + (−0.251 − 0.435i)10-s + (0.996 + 0.0838i)11-s + (0.144 − 0.249i)12-s + (0.459 − 1.41i)13-s + (0.120 − 0.696i)14-s + (0.331 + 0.241i)15-s + (−0.244 − 0.0519i)16-s + (1.18 − 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03659 - 0.487504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03659 - 0.487504i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.63 + 2.07i)T \) |
| 11 | \( 1 + (-3.30 - 0.278i)T \) |
| good | 5 | \( 1 + (1.55 + 0.330i)T + (4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (-1.65 + 5.09i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-4.88 + 5.42i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.0962 + 0.915i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.477 + 0.826i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.30 + 3.12i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.48 - 0.527i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (9.98 - 4.44i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (-2.76 + 2.01i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + (-0.309 - 2.94i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-10.4 + 2.22i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.449 - 4.28i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (9.88 + 2.10i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (6.94 + 12.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.23 - 6.89i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.182 + 1.73i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-7.03 - 7.81i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-1.40 - 4.31i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-8.07 + 13.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.23 + 6.88i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.07950222770962180377538944195, −10.11761335824233333179345004367, −9.054359714345245829083806834868, −7.73069187581057998001639910600, −7.31601555323674386905001829680, −6.22718441266725274625099449326, −5.32028636250777579100748922319, −4.09420636862520382477698263856, −3.24034154162833333436452329140, −0.69201095039564384779303389134,
1.69703213719270180030985502284, 3.58374706253440477325525187833, 4.01926540497957729255672894551, 5.56750315616372599484859376696, 6.24305645859694119295478022489, 7.30679114261524079238757857446, 8.817777008952304070520823791443, 9.404619023713491421663497447973, 10.53355793874348522372240778933, 11.32836080920075344741311847785
