| L(s) = 1 | + (−1.39 − 0.217i)2-s + (−0.276 + 1.70i)3-s + (1.90 + 0.607i)4-s + (−2.27 − 2.59i)5-s + (0.758 − 2.32i)6-s + (0.285 + 0.0375i)7-s + (−2.53 − 1.26i)8-s + (−2.84 − 0.947i)9-s + (2.61 + 4.12i)10-s + (−1.17 − 2.39i)11-s + (−1.56 + 3.08i)12-s + (1.68 + 4.97i)13-s + (−0.390 − 0.114i)14-s + (5.06 − 3.17i)15-s + (3.26 + 2.31i)16-s + (3.89 + 3.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.988 − 0.153i)2-s + (−0.159 + 0.987i)3-s + (0.952 + 0.303i)4-s + (−1.01 − 1.16i)5-s + (0.309 − 0.950i)6-s + (0.107 + 0.0141i)7-s + (−0.894 − 0.446i)8-s + (−0.948 − 0.315i)9-s + (0.827 + 1.30i)10-s + (−0.355 − 0.720i)11-s + (−0.452 + 0.891i)12-s + (0.468 + 1.38i)13-s + (−0.104 − 0.0305i)14-s + (1.30 − 0.818i)15-s + (0.815 + 0.578i)16-s + (0.945 + 0.945i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.540883 + 0.330909i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.540883 + 0.330909i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.217i)T \) |
| 3 | \( 1 + (0.276 - 1.70i)T \) |
| good | 5 | \( 1 + (2.27 + 2.59i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-0.285 - 0.0375i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (1.17 + 2.39i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (-1.68 - 4.97i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-3.89 - 3.89i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.919 + 4.62i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.838 - 6.36i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-8.30 + 0.544i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (4.64 - 2.68i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.680 - 3.41i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.56 - 11.8i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-8.04 + 3.96i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (3.01 - 0.808i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.03 - 1.54i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-1.33 - 1.51i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.482 - 7.36i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-10.8 - 5.36i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-2.12 - 5.12i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-5.18 + 12.5i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-0.386 - 1.44i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (5.44 + 4.77i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-0.978 + 0.405i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-5.54 - 3.20i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.95569557591831193185633517442, −9.839979781680829423920496031712, −9.035777949133861300110144584397, −8.498483470219101553977349249900, −7.75887783332997713182642306954, −6.39569776003666671889379615362, −5.22635236319330843686895836731, −4.14471276126586735243154737860, −3.19956429492814393485945520897, −1.10764972962148926160054551478,
0.62236220491319176463696405009, 2.43907922952318133709490526155, 3.33487900347488766652557795270, 5.39400496719442710440695456905, 6.42412173087959474934273464615, 7.27885185467842997889163927868, 7.80658727607882978730999504023, 8.360997053191939414263187155285, 9.849273050589254468073783353181, 10.70748763745246515121269169879
