| L(s) = 1 | + (−0.468 − 1.66i)3-s + (−0.102 + 0.712i)5-s + (0.0729 + 0.763i)7-s + (−2.56 + 1.56i)9-s + (0.457 − 0.183i)11-s + (3.42 + 3.59i)13-s + (1.23 − 0.162i)15-s + (6.00 − 2.07i)17-s + (1.10 + 0.105i)19-s + (1.23 − 0.479i)21-s + (−7.01 + 0.334i)23-s + (4.30 + 1.26i)25-s + (3.80 + 3.54i)27-s + (7.80 − 4.50i)29-s + (2.71 − 2.84i)31-s + ⋯ |
| L(s) = 1 | + (−0.270 − 0.962i)3-s + (−0.0457 + 0.318i)5-s + (0.0275 + 0.288i)7-s + (−0.853 + 0.520i)9-s + (0.138 − 0.0552i)11-s + (0.950 + 0.996i)13-s + (0.319 − 0.0419i)15-s + (1.45 − 0.504i)17-s + (0.254 + 0.0243i)19-s + (0.270 − 0.104i)21-s + (−1.46 + 0.0697i)23-s + (0.860 + 0.252i)25-s + (0.731 + 0.681i)27-s + (1.45 − 0.837i)29-s + (0.487 − 0.511i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43511 - 0.261603i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43511 - 0.261603i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.468 + 1.66i)T \) |
| 67 | \( 1 + (0.562 + 8.16i)T \) |
| good | 5 | \( 1 + (0.102 - 0.712i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.0729 - 0.763i)T + (-6.87 + 1.32i)T^{2} \) |
| 11 | \( 1 + (-0.457 + 0.183i)T + (7.96 - 7.59i)T^{2} \) |
| 13 | \( 1 + (-3.42 - 3.59i)T + (-0.618 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-6.00 + 2.07i)T + (13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (-1.10 - 0.105i)T + (18.6 + 3.59i)T^{2} \) |
| 23 | \( 1 + (7.01 - 0.334i)T + (22.8 - 2.18i)T^{2} \) |
| 29 | \( 1 + (-7.80 + 4.50i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.71 + 2.84i)T + (-1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-0.597 + 1.03i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.84 + 1.70i)T + (38.0 + 15.2i)T^{2} \) |
| 43 | \( 1 + (-3.66 - 3.17i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (2.65 - 5.15i)T + (-27.2 - 38.2i)T^{2} \) |
| 53 | \( 1 + (1.01 + 1.17i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (1.87 + 6.39i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (2.71 - 6.77i)T + (-44.1 - 42.0i)T^{2} \) |
| 71 | \( 1 + (-8.12 - 2.81i)T + (55.8 + 43.8i)T^{2} \) |
| 73 | \( 1 + (-10.8 - 4.34i)T + (52.8 + 50.3i)T^{2} \) |
| 79 | \( 1 + (-7.69 + 1.86i)T + (70.2 - 36.1i)T^{2} \) |
| 83 | \( 1 + (-9.04 + 11.5i)T + (-19.5 - 80.6i)T^{2} \) |
| 89 | \( 1 + (2.00 - 3.12i)T + (-36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-1.81 - 1.04i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.28233529270711477815356656031, −9.315273165976714051795480027724, −8.286866642568662564754907968556, −7.69276902929565953885476161796, −6.57958515638714386315221654564, −6.10236014841835915558126803073, −5.00463371693092053578164112938, −3.61498623222008245362990392149, −2.41938544261135611688502066154, −1.12373591916699050199020420632,
1.01742333091482602346883425682, 3.09940586428826001838601363591, 3.86671737965624015439305173308, 4.97297436458996953588496658729, 5.72971449280377383665267469676, 6.64953641695788591542880511521, 8.129756464774883953456766398710, 8.481897034992389063099506719932, 9.675131104190010938097389254628, 10.38525860771173984953671206586
