| L(s) = 1 | + (0.585 − 1.69i)2-s + (−0.458 − 0.888i)3-s + (−0.947 − 0.745i)4-s + (1.33 − 0.127i)5-s + (−1.77 + 0.254i)6-s + (2.54 + 0.728i)7-s + (1.19 − 0.769i)8-s + (−0.580 + 0.814i)9-s + (0.567 − 2.33i)10-s + (−1.82 + 0.630i)11-s + (−0.228 + 1.18i)12-s + (−1.22 − 4.16i)13-s + (2.72 − 3.87i)14-s + (−0.726 − 1.13i)15-s + (−1.16 − 4.81i)16-s + (3.70 + 1.48i)17-s + ⋯ |
| L(s) = 1 | + (0.414 − 1.19i)2-s + (−0.264 − 0.513i)3-s + (−0.473 − 0.372i)4-s + (0.598 − 0.0571i)5-s + (−0.723 + 0.104i)6-s + (0.961 + 0.275i)7-s + (0.423 − 0.271i)8-s + (−0.193 + 0.271i)9-s + (0.179 − 0.739i)10-s + (−0.549 + 0.190i)11-s + (−0.0658 + 0.341i)12-s + (−0.339 − 1.15i)13-s + (0.727 − 1.03i)14-s + (−0.187 − 0.291i)15-s + (−0.292 − 1.20i)16-s + (0.897 + 0.359i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.972937 - 1.74586i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.972937 - 1.74586i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.458 + 0.888i)T \) |
| 7 | \( 1 + (-2.54 - 0.728i)T \) |
| 23 | \( 1 + (-4.70 + 0.915i)T \) |
| good | 2 | \( 1 + (-0.585 + 1.69i)T + (-1.57 - 1.23i)T^{2} \) |
| 5 | \( 1 + (-1.33 + 0.127i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (1.82 - 0.630i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (1.22 + 4.16i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-3.70 - 1.48i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-1.10 + 0.444i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (-0.950 - 6.61i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (9.50 - 0.452i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (5.58 + 3.97i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (5.82 + 2.65i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (5.44 - 8.47i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-5.95 + 3.43i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.28 - 8.69i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (8.45 + 2.05i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (-7.28 - 3.75i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (-0.115 - 0.599i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (-8.89 - 10.2i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-9.39 + 11.9i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (1.78 - 1.86i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-4.74 - 10.3i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.715 + 15.0i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (-5.19 + 11.3i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.70479242212136886311263872989, −10.31163868178991251436567724202, −9.078044102823647864791314097911, −7.87635098920204488259792271073, −7.15265497402900346957461148632, −5.45781246262088787048865006478, −5.12444987657095433117900166101, −3.46359850601519705022451604196, −2.32956386636205746738833655473, −1.30573909688300621719278800391,
1.93260878928463686498607709026, 3.88865569888715174111884551061, 5.11636921195012771220144883406, 5.40829081264453267129396105800, 6.64271527511022256736947628904, 7.45359252561853194236371187062, 8.358460970373204695450238686069, 9.455986529818377577283496039019, 10.38269316429061822148702020250, 11.23615633015026270750474386998
