| L(s) = 1 | + (−1.33 − 0.459i)2-s + (1.57 + 1.22i)4-s + (1.74 + 0.307i)5-s + (−0.488 − 0.581i)7-s + (−1.54 − 2.36i)8-s + (−2.18 − 1.21i)10-s + (−0.557 − 3.16i)11-s + (1.31 − 0.478i)13-s + (0.385 + 1.00i)14-s + (0.978 + 3.87i)16-s + (−5.55 − 3.20i)17-s + (2.51 − 1.45i)19-s + (2.37 + 2.62i)20-s + (−0.707 + 4.48i)22-s + (−4.71 − 3.95i)23-s + ⋯ |
| L(s) = 1 | + (−0.945 − 0.324i)2-s + (0.788 + 0.614i)4-s + (0.779 + 0.137i)5-s + (−0.184 − 0.219i)7-s + (−0.546 − 0.837i)8-s + (−0.692 − 0.383i)10-s + (−0.168 − 0.953i)11-s + (0.364 − 0.132i)13-s + (0.103 + 0.267i)14-s + (0.244 + 0.969i)16-s + (−1.34 − 0.778i)17-s + (0.577 − 0.333i)19-s + (0.530 + 0.587i)20-s + (−0.150 + 0.956i)22-s + (−0.983 − 0.824i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 972 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 972 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.503046 - 0.706578i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.503046 - 0.706578i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 + 0.459i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-1.74 - 0.307i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.488 + 0.581i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.557 + 3.16i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.31 + 0.478i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (5.55 + 3.20i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.51 + 1.45i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.71 + 3.95i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.59 + 7.12i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (6.07 - 7.23i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-1.62 + 2.80i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.20 - 6.06i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-7.20 + 1.27i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.25 + 2.72i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 9.63iT - 53T^{2} \) |
| 59 | \( 1 + (0.0734 - 0.416i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.10 + 5.12i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (1.94 + 5.34i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.88 + 3.27i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.59 + 9.69i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.47 - 9.54i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (4.66 + 1.69i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.509 + 0.294i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.93 + 16.6i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−9.719143752294468114359582050656, −9.025523349709702818103732367526, −8.307517329232849632639466714260, −7.32058647148161269700025118612, −6.42740619332990384873913851319, −5.75185456520646019829674099056, −4.22816671497690162983272244962, −2.98572207854039083053747028468, −2.09827156067292543728244392994, −0.52603837902731722842796322506,
1.59360937748077628956244255319, 2.38942805697420906602914716061, 4.05598045736956630103847569212, 5.45906331290886688100391990187, 6.03279485674737358248837008038, 6.99203280973305191204367793989, 7.76656357639778633204764548459, 8.783569782523155079554251182465, 9.411593253630366208422806623190, 10.02214696425392627585984165011
