L(s) = 1 | + (0.258 − 0.965i)2-s + i·3-s + (−0.866 − 0.499i)4-s + (0.157 + 0.589i)5-s + (0.965 + 0.258i)6-s + (−0.740 − 2.53i)7-s + (−0.707 + 0.707i)8-s − 9-s + 0.610·10-s + (2.49 − 2.49i)11-s + (0.499 − 0.866i)12-s + (1.11 − 3.42i)13-s + (−2.64 + 0.0582i)14-s + (−0.589 + 0.157i)15-s + (0.500 + 0.866i)16-s + (−0.330 + 0.573i)17-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + 0.577i·3-s + (−0.433 − 0.249i)4-s + (0.0706 + 0.263i)5-s + (0.394 + 0.105i)6-s + (−0.280 − 0.959i)7-s + (−0.249 + 0.249i)8-s − 0.333·9-s + 0.192·10-s + (0.751 − 0.751i)11-s + (0.144 − 0.249i)12-s + (0.309 − 0.950i)13-s + (−0.706 + 0.0155i)14-s + (−0.152 + 0.0407i)15-s + (0.125 + 0.216i)16-s + (−0.0802 + 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08865 - 0.955919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08865 - 0.955919i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (0.740 + 2.53i)T \) |
| 13 | \( 1 + (-1.11 + 3.42i)T \) |
good | 5 | \( 1 + (-0.157 - 0.589i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.49 + 2.49i)T - 11iT^{2} \) |
| 17 | \( 1 + (0.330 - 0.573i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.30 - 1.30i)T - 19iT^{2} \) |
| 23 | \( 1 + (-5.75 + 3.32i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.81 + 4.86i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.17 + 0.583i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-2.18 - 0.584i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (1.83 + 6.84i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (6.91 - 3.99i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.42 - 1.18i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.24 - 9.09i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-9.32 + 2.49i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 - 13.1iT - 61T^{2} \) |
| 67 | \( 1 + (10.1 + 10.1i)T + 67iT^{2} \) |
| 71 | \( 1 + (-3.44 + 12.8i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (1.08 - 4.06i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (3.20 - 5.55i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.17 - 5.17i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.90 - 14.5i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (3.44 + 0.924i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.63577648496025763212185589345, −10.03522668191162318226274623491, −8.987469525769157900073645519852, −8.182360675389983908909200779493, −6.81917978347022054076006981913, −5.90280473326265653219971435679, −4.65029008801781169770134149514, −3.72098372970895739607205555511, −2.87319521813406055192605165762, −0.869918465866741661487758416504,
1.66331471287903753837642313542, 3.20579060691366140839917273611, 4.65083015429429311984668151891, 5.52173763334491857658260642562, 6.75754668104600845319569096326, 6.97372696232067274269266073247, 8.497551724888251756977366487926, 8.965108756383692343293265100021, 9.775628547093418361917807309790, 11.33902304675476111486456205598