| L(s) = 1 | + (0.123 + 2.59i)2-s + (−4.72 + 0.450i)4-s + (−1.67 − 1.93i)5-s + (−0.534 + 1.03i)7-s + (−1.01 − 7.04i)8-s + (4.80 − 4.58i)10-s + (0.857 − 2.47i)11-s + (2.20 − 2.80i)13-s + (−2.75 − 1.25i)14-s + (8.84 − 1.70i)16-s + (−0.0350 + 0.366i)17-s + (5.78 − 2.98i)19-s + (8.77 + 8.36i)20-s + (6.53 + 1.91i)22-s + (−6.88 − 1.67i)23-s + ⋯ |
| L(s) = 1 | + (0.0873 + 1.83i)2-s + (−2.36 + 0.225i)4-s + (−0.748 − 0.863i)5-s + (−0.202 + 0.391i)7-s + (−0.358 − 2.49i)8-s + (1.51 − 1.44i)10-s + (0.258 − 0.747i)11-s + (0.611 − 0.777i)13-s + (−0.736 − 0.336i)14-s + (2.21 − 0.426i)16-s + (−0.00849 + 0.0889i)17-s + (1.32 − 0.683i)19-s + (1.96 + 1.87i)20-s + (1.39 + 0.409i)22-s + (−1.43 − 0.348i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.844146 + 0.178236i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.844146 + 0.178236i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 67 | \( 1 + (4.14 - 7.05i)T \) |
| good | 2 | \( 1 + (-0.123 - 2.59i)T + (-1.99 + 0.190i)T^{2} \) |
| 5 | \( 1 + (1.67 + 1.93i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (0.534 - 1.03i)T + (-4.06 - 5.70i)T^{2} \) |
| 11 | \( 1 + (-0.857 + 2.47i)T + (-8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (-2.20 + 2.80i)T + (-3.06 - 12.6i)T^{2} \) |
| 17 | \( 1 + (0.0350 - 0.366i)T + (-16.6 - 3.21i)T^{2} \) |
| 19 | \( 1 + (-5.78 + 2.98i)T + (11.0 - 15.4i)T^{2} \) |
| 23 | \( 1 + (6.88 + 1.67i)T + (20.4 + 10.5i)T^{2} \) |
| 29 | \( 1 + (6.37 - 3.67i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.12 - 1.43i)T + (-7.30 + 30.1i)T^{2} \) |
| 37 | \( 1 + (-1.36 + 2.35i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.31 + 8.86i)T + (-13.4 - 38.7i)T^{2} \) |
| 43 | \( 1 + (-1.73 + 0.792i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (-8.06 + 8.46i)T + (-2.23 - 46.9i)T^{2} \) |
| 53 | \( 1 + (-0.0162 + 0.0355i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (8.61 - 1.23i)T + (56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (2.28 - 0.790i)T + (47.9 - 37.7i)T^{2} \) |
| 71 | \( 1 + (1.13 + 11.8i)T + (-69.7 + 13.4i)T^{2} \) |
| 73 | \( 1 + (3.21 + 9.30i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (4.66 + 11.6i)T + (-57.1 + 54.5i)T^{2} \) |
| 83 | \( 1 + (2.80 + 14.5i)T + (-77.0 + 30.8i)T^{2} \) |
| 89 | \( 1 + (0.420 - 1.43i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (6.65 + 3.84i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.54437334367407957968193491736, −9.068144971650852789280139948979, −8.892507664904535998841820286639, −7.87862540368122971461260921625, −7.34478899041520897699679806842, −5.98549562565507770290705910642, −5.57272618930501883384864779283, −4.45181762429632158543359326899, −3.50623492534602421089628239556, −0.51973883834808682756262564557,
1.45034991360550245572240636366, 2.76895511329663106693101992692, 3.84356492358677584306277237606, 4.25063669086742311580970108104, 5.85013645224001288318834559466, 7.24211578914497203740511201403, 8.085258568172321576418637279620, 9.533208589887502032430506497605, 9.753801085402376794891085446400, 10.86008362870208653821975528385
