L(s) = 1 | + (0.456 − 1.40i)2-s + (0.809 − 0.587i)3-s + (−0.147 − 0.107i)4-s + (−0.456 − 1.40i)6-s + (−1.85 − 1.34i)7-s + (2.17 − 1.57i)8-s + (0.309 − 0.951i)9-s + (−3.12 − 1.12i)11-s − 0.182·12-s + (0.661 − 2.03i)13-s + (−2.74 + 1.99i)14-s + (−1.33 − 4.11i)16-s + (−0.168 − 0.517i)17-s + (−1.19 − 0.868i)18-s + (1.76 − 1.28i)19-s + ⋯ |
L(s) = 1 | + (0.322 − 0.993i)2-s + (0.467 − 0.339i)3-s + (−0.0737 − 0.0535i)4-s + (−0.186 − 0.573i)6-s + (−0.701 − 0.509i)7-s + (0.768 − 0.558i)8-s + (0.103 − 0.317i)9-s + (−0.940 − 0.339i)11-s − 0.0526·12-s + (0.183 − 0.564i)13-s + (−0.733 + 0.532i)14-s + (−0.334 − 1.02i)16-s + (−0.0408 − 0.125i)17-s + (−0.281 − 0.204i)18-s + (0.405 − 0.294i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.645910 - 1.94144i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.645910 - 1.94144i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (3.12 + 1.12i)T \) |
good | 2 | \( 1 + (-0.456 + 1.40i)T + (-1.61 - 1.17i)T^{2} \) |
| 7 | \( 1 + (1.85 + 1.34i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-0.661 + 2.03i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.168 + 0.517i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.76 + 1.28i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 2.03T + 23T^{2} \) |
| 29 | \( 1 + (-8.04 - 5.84i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.09 + 6.44i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (7.13 + 5.18i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.47 + 1.07i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 0.620T + 43T^{2} \) |
| 47 | \( 1 + (0.305 - 0.222i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (3.58 - 11.0i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.53 - 4.74i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.69 - 8.29i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 9.75T + 67T^{2} \) |
| 71 | \( 1 + (4.63 + 14.2i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.35 - 4.61i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.85 - 8.77i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.92 - 8.98i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 0.583T + 89T^{2} \) |
| 97 | \( 1 + (-1.66 + 5.11i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.26419349600828346398048145044, −9.223977122823830907997769224319, −8.124812702440468426330950971299, −7.38083125831503882968615971753, −6.53804901297656448338894472098, −5.27074035175478966212826103650, −4.01634945441813284811754735551, −3.14802954771315114921889526962, −2.42525252715385162529797678767, −0.855701841355085177764351660716,
2.04804665410323933848652577283, 3.19962274974669881913168687593, 4.53889152798520646499557379583, 5.30702358912006697899771044787, 6.30582358035172574166369532025, 6.95817208423389544886981626623, 8.054785030955556006430734044914, 8.549371126153607707844767650760, 9.809848038956683892158076538268, 10.24871841223551945032433560255