L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (3.41 − 1.10i)5-s + (0.587 − 0.809i)7-s + (−0.809 + 0.587i)8-s − 3.58i·10-s + (−3.26 + 0.582i)11-s + (2.13 + 0.692i)13-s + (−0.587 − 0.809i)14-s + (0.309 + 0.951i)16-s + (−0.427 − 1.31i)17-s + (−3.86 − 5.32i)19-s + (−3.41 − 1.10i)20-s + (−0.455 + 3.28i)22-s − 7.59i·23-s + ⋯ |
L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (1.52 − 0.495i)5-s + (0.222 − 0.305i)7-s + (−0.286 + 0.207i)8-s − 1.13i·10-s + (−0.984 + 0.175i)11-s + (0.590 + 0.191i)13-s + (−0.157 − 0.216i)14-s + (0.0772 + 0.237i)16-s + (−0.103 − 0.319i)17-s + (−0.887 − 1.22i)19-s + (−0.762 − 0.247i)20-s + (−0.0970 + 0.700i)22-s − 1.58i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.189108565\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.189108565\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.587 + 0.809i)T \) |
| 11 | \( 1 + (3.26 - 0.582i)T \) |
good | 5 | \( 1 + (-3.41 + 1.10i)T + (4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (-2.13 - 0.692i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.427 + 1.31i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (3.86 + 5.32i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 7.59iT - 23T^{2} \) |
| 29 | \( 1 + (0.0255 + 0.0185i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.52 + 4.68i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.90 - 5.01i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-6.00 + 4.36i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 12.2iT - 43T^{2} \) |
| 47 | \( 1 + (-0.0507 - 0.0698i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.10 + 0.683i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.98 + 2.73i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.970 - 0.315i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + (-14.6 + 4.76i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.79 - 5.22i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.16 - 1.35i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (3.14 + 9.69i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 5.35iT - 89T^{2} \) |
| 97 | \( 1 + (-0.615 + 1.89i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.423681817700297761989649022672, −8.778583557335353943570261838154, −7.897292197373978254053864195197, −6.53429501480231819891572180339, −5.97100624391458579038285491840, −4.86426967221888984046379007183, −4.45082225438397070846636890661, −2.74712345440146025448588782323, −2.15719988398684467245932077450, −0.835030671545909678087970029774,
1.68068762105722609593755311333, 2.71932831644120745295208729764, 3.86718477923162750230103766171, 5.21771877080350727742138290278, 5.80415263849211846740224194617, 6.26322597949462190424225085426, 7.36342752571808379561004535261, 8.171007664198780364386389420813, 8.979902871717893062675106132423, 9.812638196010133737663996016061