| L(s) = 1 | + (−1.02 − 2.61i)2-s + (1.47 − 0.910i)3-s + (−4.29 + 3.98i)4-s + (1.33 + 0.640i)5-s + (−3.88 − 2.91i)6-s + (0.948 + 2.46i)7-s + (9.76 + 4.70i)8-s + (1.34 − 2.68i)9-s + (0.309 − 4.12i)10-s + (2.77 − 3.48i)11-s + (−2.70 + 9.79i)12-s + (1.57 + 4.02i)13-s + (5.47 − 5.00i)14-s + (2.54 − 0.266i)15-s + (1.39 − 18.6i)16-s + (−1.37 − 1.27i)17-s + ⋯ |
| L(s) = 1 | + (−0.724 − 1.84i)2-s + (0.850 − 0.525i)3-s + (−2.14 + 1.99i)4-s + (0.594 + 0.286i)5-s + (−1.58 − 1.18i)6-s + (0.358 + 0.933i)7-s + (3.45 + 1.66i)8-s + (0.447 − 0.894i)9-s + (0.0978 − 1.30i)10-s + (0.838 − 1.05i)11-s + (−0.780 + 2.82i)12-s + (0.437 + 1.11i)13-s + (1.46 − 1.33i)14-s + (0.656 − 0.0688i)15-s + (0.348 − 4.65i)16-s + (−0.333 − 0.309i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.692312 - 1.17665i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.692312 - 1.17665i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.47 + 0.910i)T \) |
| 7 | \( 1 + (-0.948 - 2.46i)T \) |
| good | 2 | \( 1 + (1.02 + 2.61i)T + (-1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (-1.33 - 0.640i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (-2.77 + 3.48i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.57 - 4.02i)T + (-9.52 + 8.84i)T^{2} \) |
| 17 | \( 1 + (1.37 + 1.27i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.601 + 1.04i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.20 - 5.29i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-1.85 + 0.571i)T + (23.9 - 16.3i)T^{2} \) |
| 31 | \( 1 + (0.861 + 1.49i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.63 + 1.43i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-0.243 + 0.166i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (-3.85 - 2.62i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (-1.75 - 4.46i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (-0.863 - 0.266i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (8.97 + 6.12i)T + (21.5 + 54.9i)T^{2} \) |
| 61 | \( 1 + (6.96 + 6.46i)T + (4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (0.615 + 1.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.485 + 2.12i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (11.5 - 1.74i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (6.42 - 11.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.22 + 8.20i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-1.90 + 4.85i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-4.61 - 7.99i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.06459489766089558085222861992, −9.662241447593421597238702236211, −9.158742028989702583640516446622, −8.655627324133698011240933547692, −7.66402350840822189634324309921, −6.17358327618889014972845866519, −4.36612078236221283399677832116, −3.25917860724575552360973300547, −2.30747769516625853265834159398, −1.38997200224983504165369272160,
1.41799831197561569547848296664, 4.09759327356059656094655640333, 4.79616585463588055087484538320, 5.95601302929402734171018261564, 7.05431244826318621377422976247, 7.76196817109856917951141359730, 8.630255185014700436326233636553, 9.283727063875839588694042177751, 10.20598106552895263636561128321, 10.57681327297680466686894174874
