| L(s) = 1 | + (0.0474 − 0.0373i)2-s + (0.814 − 0.580i)3-s + (−0.470 + 1.94i)4-s + (−2.52 − 0.486i)5-s + (0.0170 − 0.0579i)6-s + (−2.47 − 0.925i)7-s + (0.100 + 0.219i)8-s + (0.327 − 0.945i)9-s + (−0.137 + 0.0711i)10-s + (−0.645 + 0.821i)11-s + (0.741 + 1.85i)12-s + (2.60 − 4.05i)13-s + (−0.152 + 0.0486i)14-s + (−2.33 + 1.06i)15-s + (−3.53 − 1.82i)16-s + (−4.48 + 4.27i)17-s + ⋯ |
| L(s) = 1 | + (0.0335 − 0.0264i)2-s + (0.470 − 0.334i)3-s + (−0.235 + 0.970i)4-s + (−1.12 − 0.217i)5-s + (0.00694 − 0.0236i)6-s + (−0.936 − 0.349i)7-s + (0.0354 + 0.0776i)8-s + (0.109 − 0.315i)9-s + (−0.0436 + 0.0224i)10-s + (−0.194 + 0.247i)11-s + (0.214 + 0.535i)12-s + (0.723 − 1.12i)13-s + (−0.0407 + 0.0129i)14-s + (−0.603 + 0.275i)15-s + (−0.883 − 0.455i)16-s + (−1.08 + 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0208763 - 0.152036i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0208763 - 0.152036i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.47 + 0.925i)T \) |
| 23 | \( 1 + (3.72 - 3.01i)T \) |
| good | 2 | \( 1 + (-0.0474 + 0.0373i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (2.52 + 0.486i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (0.645 - 0.821i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-2.60 + 4.05i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.48 - 4.27i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (4.83 + 4.61i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (8.05 + 2.36i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.114 + 1.19i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-8.56 - 2.96i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-3.91 - 3.39i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (1.99 + 0.910i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-8.34 + 4.82i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.34 - 0.159i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (1.83 + 3.55i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (0.0744 - 0.104i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (5.63 - 14.0i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-1.05 - 7.30i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (2.26 + 0.549i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-9.43 - 0.449i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (4.05 + 4.67i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-6.70 + 0.639i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (-0.255 + 0.294i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.73991531410529761964157889338, −9.452878017104888087749716791804, −8.523104960144349363528445535227, −7.948385988843385315552662853053, −7.17941394379409838780421445765, −6.09917595961836004757902590511, −4.21237799448165606626027253018, −3.79775780447943655334380253306, −2.60197062340514694722510684802, −0.082212486401965811089185643435,
2.24176242085756334743189209890, 3.78385806528353387836649885133, 4.41463247562476943773826345147, 5.91618999993411108291956764929, 6.67680519058742476975961217603, 7.85091234859503464994695324798, 9.029217801657738590980214214160, 9.344269539506872502283207279448, 10.62177881577356062338535973054, 11.13347689166230152691432421486
