L(s) = 1 | + 1.40·2-s + 1.73i·3-s − 0.0315·4-s + 2.23i·5-s + 2.43i·6-s − 2.85·8-s − 2.99·9-s + 3.13i·10-s − 0.0545i·12-s − 3.87·15-s − 3.93·16-s + 8.06i·17-s − 4.20·18-s − 6.65i·19-s − 0.0704i·20-s + ⋯ |
L(s) = 1 | + 0.992·2-s + 0.999i·3-s − 0.0157·4-s + 0.999i·5-s + 0.992i·6-s − 1.00·8-s − 0.999·9-s + 0.992i·10-s − 0.0157i·12-s − 0.999·15-s − 0.983·16-s + 1.95i·17-s − 0.992·18-s − 1.52i·19-s − 0.0157i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.328344 + 1.53494i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.328344 + 1.53494i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.40T + 2T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 8.06iT - 17T^{2} \) |
| 19 | \( 1 + 6.65iT - 19T^{2} \) |
| 23 | \( 1 + 0.222T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 10.2iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 1.02iT - 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.0iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 5.83T + 79T^{2} \) |
| 83 | \( 1 - 15.0iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.71542859671847369993902556863, −10.15223989480883181723147859120, −9.050759868501892042108359337024, −8.404060501191915100053070792617, −6.93167939795164778976083005701, −6.08538487841243942014333313456, −5.24816055392908158528127673490, −4.23739946350880805180450597500, −3.51607947712817406394301239782, −2.59901605983030926558498687515,
0.57302287658251904550117412076, 2.21695490357010426479938978783, 3.51161548464980265349733198777, 4.65168799883800504381330877533, 5.50366773787512812288121069515, 6.15461513236559358525089219886, 7.38507715002068529375761176244, 8.182230874784833754920877293292, 9.092549660547251694777629423602, 9.791005643968569164291772531069