| L(s) = 1 | + (2.01 − 1.29i)2-s + (1.55 − 3.41i)4-s + (−1.75 − 2.02i)5-s + (−0.938 + 0.603i)7-s + (−0.597 − 4.15i)8-s + (−6.17 − 1.81i)10-s + (−2.74 − 3.16i)11-s + (0.736 − 5.12i)13-s + (−1.11 + 2.43i)14-s + (−1.67 − 1.93i)16-s + (3.20 + 7.00i)17-s + (0.207 + 0.133i)19-s + (−9.65 + 2.83i)20-s + (−9.63 − 2.82i)22-s + (2.78 − 0.818i)23-s + ⋯ |
| L(s) = 1 | + (1.42 − 0.916i)2-s + (0.778 − 1.70i)4-s + (−0.785 − 0.906i)5-s + (−0.354 + 0.227i)7-s + (−0.211 − 1.46i)8-s + (−1.95 − 0.573i)10-s + (−0.827 − 0.954i)11-s + (0.204 − 1.42i)13-s + (−0.296 + 0.650i)14-s + (−0.419 − 0.483i)16-s + (0.776 + 1.69i)17-s + (0.0476 + 0.0305i)19-s + (−2.15 + 0.633i)20-s + (−2.05 − 0.603i)22-s + (0.581 − 0.170i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.779203 - 2.40105i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.779203 - 2.40105i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 67 | \( 1 + (-7.52 - 3.22i)T \) |
| good | 2 | \( 1 + (-2.01 + 1.29i)T + (0.830 - 1.81i)T^{2} \) |
| 5 | \( 1 + (1.75 + 2.02i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (0.938 - 0.603i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (2.74 + 3.16i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.736 + 5.12i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-3.20 - 7.00i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.207 - 0.133i)T + (7.89 + 17.2i)T^{2} \) |
| 23 | \( 1 + (-2.78 + 0.818i)T + (19.3 - 12.4i)T^{2} \) |
| 29 | \( 1 - 0.379T + 29T^{2} \) |
| 31 | \( 1 + (1.27 + 8.83i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 - 5.05T + 37T^{2} \) |
| 41 | \( 1 + (-0.891 - 1.95i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.84 - 6.22i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-6.36 + 1.86i)T + (39.5 - 25.4i)T^{2} \) |
| 53 | \( 1 + (2.32 - 5.08i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (1.02 + 7.12i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-0.986 + 1.13i)T + (-8.68 - 60.3i)T^{2} \) |
| 71 | \( 1 + (-1.11 + 2.44i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (5.12 - 5.91i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (0.725 - 5.04i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-2.54 - 2.93i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (12.5 + 3.68i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + 6.90T + 97T^{2} \) |
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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
−10.75701472401086344141116910729, −9.787364922783961309314839781900, −8.340698102651351240653056125907, −7.912477943995638013407910671644, −5.95993287342267905611397498731, −5.63105574860325228724040610681, −4.47826068371621288965736083269, −3.58949016611154597947017899553, −2.74013911355282468530920458822, −0.929493408372804059500641295079,
2.70003766971037348830941518443, 3.58887360803093043247024618245, 4.57609526105083910380940744913, 5.38050087189527186891910866893, 6.76273274814103146655765732611, 7.11136350586182119732977296889, 7.70557626667353663797790635123, 9.181163515223466020602788765400, 10.27655934396916567119837648910, 11.37782906080144638900724948181
