| L(s) = 1 | + (1.83 − 1.54i)2-s + (0.654 − 3.71i)4-s + (0.0874 − 0.0318i)5-s + (−0.100 − 0.571i)7-s + (−2.12 − 3.67i)8-s + (0.111 − 0.193i)10-s + (−2.90 − 1.05i)11-s + (3.21 + 2.70i)13-s + (−1.06 − 0.895i)14-s + (−2.50 − 0.909i)16-s + (−0.995 + 1.72i)17-s + (1.92 + 3.33i)19-s + (−0.0609 − 0.345i)20-s + (−6.98 + 2.54i)22-s + (0.773 − 4.38i)23-s + ⋯ |
| L(s) = 1 | + (1.30 − 1.09i)2-s + (0.327 − 1.85i)4-s + (0.0391 − 0.0142i)5-s + (−0.0380 − 0.215i)7-s + (−0.750 − 1.29i)8-s + (0.0353 − 0.0612i)10-s + (−0.876 − 0.318i)11-s + (0.892 + 0.749i)13-s + (−0.285 − 0.239i)14-s + (−0.625 − 0.227i)16-s + (−0.241 + 0.418i)17-s + (0.441 + 0.764i)19-s + (−0.0136 − 0.0772i)20-s + (−1.48 + 0.541i)22-s + (0.161 − 0.914i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58377 - 1.68751i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58377 - 1.68751i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-1.83 + 1.54i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (-0.0874 + 0.0318i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.100 + 0.571i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (2.90 + 1.05i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.21 - 2.70i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.995 - 1.72i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.92 - 3.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.773 + 4.38i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (4.90 - 4.11i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.287 + 1.63i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.01 - 3.49i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.839 - 0.704i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-6.48 - 2.36i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.623 + 3.53i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 5.40T + 53T^{2} \) |
| 59 | \( 1 + (9.66 - 3.51i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (2.29 + 12.9i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (6.76 + 5.68i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.572 + 0.991i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.0977 + 0.169i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.52 - 4.63i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-11.4 + 9.57i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (0.776 + 1.34i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.97 + 1.81i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.95016717008813257755134593965, −11.00647658722442011731455492461, −10.49355067395804552497432296895, −9.256564376238320703840360791458, −7.891611794529523281368093707547, −6.33179947708599720364809280803, −5.40033776720884442697299309525, −4.20859885297364881836559813022, −3.23019857450440572661328996300, −1.74669830717451026072632167227,
2.84990715226984052934780588609, 4.13628936297831052923226353323, 5.34041696829377863088992124181, 5.98723225058324055242164454936, 7.27854691541108743408572754277, 7.938634612663545938530712341201, 9.247882372266761254217955730539, 10.63398218401960101156650856239, 11.74983828274093300407249966909, 12.74973202770893019148143210217
