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
−9.812638196010133737663996016061, −8.979902871717893062675106132423, −8.171007664198780364386389420813, −7.36342752571808379561004535261, −6.26322597949462190424225085426, −5.80415263849211846740224194617, −5.21771877080350727742138290278, −3.86718477923162750230103766171, −2.71932831644120745295208729764, −1.68068762105722609593755311333,
0.835030671545909678087970029774, 2.15719988398684467245932077450, 2.74712345440146025448588782323, 4.45082225438397070846636890661, 4.86426967221888984046379007183, 5.97100624391458579038285491840, 6.53429501480231819891572180339, 7.897292197373978254053864195197, 8.778583557335353943570261838154, 9.423681817700297761989649022672