| L(s) = 1 | + (−0.781 − 0.623i)2-s + (0.744 − 1.56i)3-s + (0.222 + 0.974i)4-s + (−0.0165 − 0.220i)5-s + (−1.55 + 0.758i)6-s + (1.78 − 1.95i)7-s + (0.433 − 0.900i)8-s + (−1.89 − 2.32i)9-s + (−0.124 + 0.182i)10-s + (−4.87 + 1.91i)11-s + (1.69 + 0.378i)12-s + (−2.36 + 0.928i)13-s + (−2.61 + 0.419i)14-s + (−0.356 − 0.138i)15-s + (−0.900 + 0.433i)16-s + (−2.96 + 0.915i)17-s + ⋯ |
| L(s) = 1 | + (−0.552 − 0.440i)2-s + (0.429 − 0.902i)3-s + (0.111 + 0.487i)4-s + (−0.00738 − 0.0985i)5-s + (−0.635 + 0.309i)6-s + (0.673 − 0.739i)7-s + (0.153 − 0.318i)8-s + (−0.630 − 0.776i)9-s + (−0.0393 + 0.0577i)10-s + (−1.46 + 0.576i)11-s + (0.487 + 0.109i)12-s + (−0.656 + 0.257i)13-s + (−0.698 + 0.112i)14-s + (−0.0921 − 0.0357i)15-s + (−0.225 + 0.108i)16-s + (−0.720 + 0.222i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.122269 + 0.645549i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.122269 + 0.645549i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.781 + 0.623i)T \) |
| 3 | \( 1 + (-0.744 + 1.56i)T \) |
| 7 | \( 1 + (-1.78 + 1.95i)T \) |
| good | 5 | \( 1 + (0.0165 + 0.220i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (4.87 - 1.91i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (2.36 - 0.928i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (2.96 - 0.915i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (3.20 + 1.85i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.35 + 2.53i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.973 - 3.15i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + 5.27iT - 31T^{2} \) |
| 37 | \( 1 + (6.52 + 6.05i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-5.70 + 3.89i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (-1.33 - 0.907i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (2.23 - 2.80i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (6.75 + 7.27i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (4.88 - 2.35i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-5.48 - 1.25i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 - 1.83T + 67T^{2} \) |
| 71 | \( 1 + (2.90 - 0.662i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-5.28 - 2.07i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 - 9.48T + 79T^{2} \) |
| 83 | \( 1 + (4.75 - 12.1i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-15.5 + 2.34i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (8.71 - 5.03i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.632028233827333948843402587814, −8.670378123427453807809622088576, −8.002818582477806563042536447848, −7.32761030976821073409907692914, −6.62069841989755452538726412808, −5.12042664889305495753509473616, −4.12469741348220782204483986849, −2.61983080260728451322333589271, −1.93445671176995882136732214551, −0.32744139852754176615331075549,
2.17669985224730874777959747846, 3.08087056930426404402731473331, 4.71179660933671982398722195012, 5.22117098128903931276333935334, 6.20195864073037668033143740423, 7.58834894599092740664332771789, 8.234130091937328559841892661271, 8.791252541156485244900322981343, 9.695981002654433025309598696038, 10.57226538673088050804968238318
