| L(s) = 1 | + (−1.93 + 1.52i)2-s + (0.814 − 0.580i)3-s + (0.955 − 3.93i)4-s + (2.26 + 0.437i)5-s + (−0.693 + 2.36i)6-s + (2.27 + 1.35i)7-s + (2.09 + 4.59i)8-s + (0.327 − 0.945i)9-s + (−5.05 + 2.60i)10-s + (−3.37 + 4.29i)11-s + (−1.50 − 3.76i)12-s + (1.63 − 2.53i)13-s + (−6.45 + 0.847i)14-s + (2.10 − 0.959i)15-s + (−3.84 − 1.98i)16-s + (2.54 − 2.42i)17-s + ⋯ |
| L(s) = 1 | + (−1.36 + 1.07i)2-s + (0.470 − 0.334i)3-s + (0.477 − 1.96i)4-s + (1.01 + 0.195i)5-s + (−0.282 + 0.963i)6-s + (0.859 + 0.510i)7-s + (0.742 + 1.62i)8-s + (0.109 − 0.315i)9-s + (−1.59 + 0.823i)10-s + (−1.01 + 1.29i)11-s + (−0.434 − 1.08i)12-s + (0.452 − 0.704i)13-s + (−1.72 + 0.226i)14-s + (0.542 − 0.247i)15-s + (−0.961 − 0.495i)16-s + (0.617 − 0.589i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.869256 + 0.632856i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.869256 + 0.632856i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.814 + 0.580i)T \) |
| 7 | \( 1 + (-2.27 - 1.35i)T \) |
| 23 | \( 1 + (-1.29 - 4.61i)T \) |
| good | 2 | \( 1 + (1.93 - 1.52i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-2.26 - 0.437i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (3.37 - 4.29i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.63 + 2.53i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-2.54 + 2.42i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.307 + 0.293i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (-8.46 - 2.48i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.321 + 3.36i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-4.10 - 1.41i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (6.43 + 5.57i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.852 + 0.389i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (3.54 - 2.04i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.08 + 0.242i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (2.72 + 5.28i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (3.60 - 5.05i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (1.94 - 4.85i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-1.96 - 13.6i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (7.99 + 1.94i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-14.1 - 0.671i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-1.19 - 1.38i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-7.77 + 0.742i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (-0.0813 + 0.0938i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.56921159064703563612022878906, −9.984706097653547847041184991319, −9.277019195013340241756482279828, −8.291566036871709708153631818252, −7.71905437202145058570665175220, −6.88958682540236596383277006411, −5.77254983200121814234305983240, −5.06808973518322293835467829467, −2.57363545819081804414186968751, −1.44778432520541135811499947150,
1.16448568908613193386004912739, 2.30899139050632980394229866031, 3.41745206710287475733879186443, 4.88340533369430346767249814408, 6.27542164820214927559644024751, 7.85369028588824419972521531238, 8.380593352762056934469735302264, 9.047086023506057076914779224563, 10.13330081254776754272746866768, 10.51113244258827462737681514543
