| L(s) = 1 | + (−0.725 − 0.194i)2-s + (2.68 − 1.34i)3-s + (−2.97 − 1.71i)4-s + (4.81 − 1.33i)5-s + (−2.20 + 0.451i)6-s + (−1.79 − 0.481i)7-s + (3.94 + 3.94i)8-s + (5.40 − 7.19i)9-s + (−3.75 + 0.0302i)10-s + (5.82 + 10.0i)11-s + (−10.2 − 0.617i)12-s + (−19.8 + 5.30i)13-s + (1.21 + 0.698i)14-s + (11.1 − 10.0i)15-s + (4.77 + 8.26i)16-s + (−10.0 + 10.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.362 − 0.0972i)2-s + (0.894 − 0.447i)3-s + (−0.743 − 0.429i)4-s + (0.963 − 0.266i)5-s + (−0.368 + 0.0753i)6-s + (−0.256 − 0.0687i)7-s + (0.493 + 0.493i)8-s + (0.600 − 0.799i)9-s + (−0.375 + 0.00302i)10-s + (0.529 + 0.916i)11-s + (−0.857 − 0.0514i)12-s + (−1.52 + 0.408i)13-s + (0.0864 + 0.0499i)14-s + (0.742 − 0.669i)15-s + (0.298 + 0.516i)16-s + (−0.589 + 0.589i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.05094 - 0.415324i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05094 - 0.415324i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.68 + 1.34i)T \) |
| 5 | \( 1 + (-4.81 + 1.33i)T \) |
| good | 2 | \( 1 + (0.725 + 0.194i)T + (3.46 + 2i)T^{2} \) |
| 7 | \( 1 + (1.79 + 0.481i)T + (42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (-5.82 - 10.0i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (19.8 - 5.30i)T + (146. - 84.5i)T^{2} \) |
| 17 | \( 1 + (10.0 - 10.0i)T - 289iT^{2} \) |
| 19 | \( 1 - 10.8iT - 361T^{2} \) |
| 23 | \( 1 + (1.34 - 0.360i)T + (458. - 264.5i)T^{2} \) |
| 29 | \( 1 + (-20.7 + 12.0i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-21.6 + 37.4i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (32.5 - 32.5i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + (20.5 - 35.5i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-2.14 + 8.01i)T + (-1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (-17.2 - 4.62i)T + (1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (51.3 + 51.3i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-24.3 - 14.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (41.1 + 71.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-8.65 - 32.3i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 99.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + (22.3 + 22.3i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + (52.9 - 30.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-13.3 + 49.9i)T + (-5.96e3 - 3.44e3i)T^{2} \) |
| 89 | \( 1 - 113. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (29.5 + 7.92i)T + (8.14e3 + 4.70e3i)T^{2} \) |
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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
−15.04081099918070357650906821987, −14.26128058953737332248645313311, −13.32423032463677154716328320107, −12.30146814598244990500851952272, −9.911909935627411482792796048115, −9.596161548403905921985705892599, −8.257704484665506221348596586143, −6.59202678497153019674044431455, −4.60418860635342873203345763169, −1.93612092645907280523258923379,
2.97907235949605212522800150586, 4.91067896252859586161554198812, 7.11305565230880167708400489337, 8.687640362031736288132981974596, 9.453628778264439721182185630932, 10.46258946527338284237823048508, 12.56518639028710236261011335098, 13.77165351502571155401288739283, 14.28248185773843680958323974474, 15.79598113595151056339764887580
