L(s) = 1 | + (−1.29 + 2.23i)3-s + (−0.5 − 0.866i)5-s + (−0.292 + 2.62i)7-s + (−1.83 − 3.18i)9-s + (−0.839 + 1.45i)11-s − 4.84·13-s + 2.58·15-s + (−1 + 1.73i)17-s + (−3.42 − 5.92i)19-s + (−5.50 − 4.05i)21-s + (1.13 + 1.95i)23-s + (−0.499 + 0.866i)25-s + 1.75·27-s + 3.32·29-s + (−4.58 + 7.94i)31-s + ⋯ |
L(s) = 1 | + (−0.746 + 1.29i)3-s + (−0.223 − 0.387i)5-s + (−0.110 + 0.993i)7-s + (−0.613 − 1.06i)9-s + (−0.253 + 0.438i)11-s − 1.34·13-s + 0.667·15-s + (−0.242 + 0.420i)17-s + (−0.785 − 1.36i)19-s + (−1.20 − 0.884i)21-s + (0.235 + 0.408i)23-s + (−0.0999 + 0.173i)25-s + 0.337·27-s + 0.616·29-s + (−0.823 + 1.42i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0480615 + 0.551008i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0480615 + 0.551008i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.292 - 2.62i)T \) |
good | 3 | \( 1 + (1.29 - 2.23i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (0.839 - 1.45i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.42 + 5.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.13 - 1.95i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 + (4.58 - 7.94i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.42 - 2.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 9.52T + 41T^{2} \) |
| 43 | \( 1 - 6.58T + 43T^{2} \) |
| 47 | \( 1 + (-6.10 - 10.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.74 - 6.48i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.24 - 5.62i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.87 + 4.98i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-5.84 + 10.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.84 + 4.93i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + (-2.92 - 5.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−12.26935787138324457670064917027, −11.22215599826273819961369679950, −10.50048207077455719484642455523, −9.414425640575757529453898702209, −8.927216585895992864730906058665, −7.42612207343484028981205389937, −6.02983673712212829535278577653, −4.98945116442219325126059479041, −4.43074120077593723920974351735, −2.68089200758246309366403551718,
0.43678025571468972624121645034, 2.29818703879983319749571464018, 4.09076195870915743930351395785, 5.59150258073620404340717824829, 6.62543321457133270199307982323, 7.37600063901439103038055379399, 8.047109763001687147315687759710, 9.721985932483798865040004331826, 10.72725020271821621800707018053, 11.45600439630618942355857030118