L(s) = 1 | + 2.32i·3-s + 4.13i·5-s + 2.86i·7-s − 2.42·9-s − 0.105i·11-s + 4.79·13-s − 9.63·15-s + (−2.80 + 3.02i)17-s + 6.25·19-s − 6.66·21-s + 3.72i·23-s − 12.1·25-s + 1.34i·27-s + 0.0883i·29-s − 5.14i·31-s + ⋯ |
L(s) = 1 | + 1.34i·3-s + 1.84i·5-s + 1.08i·7-s − 0.807·9-s − 0.0318i·11-s + 1.32·13-s − 2.48·15-s + (−0.679 + 0.733i)17-s + 1.43·19-s − 1.45·21-s + 0.776i·23-s − 2.42·25-s + 0.258i·27-s + 0.0164i·29-s − 0.923i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.893060759\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.893060759\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (2.80 - 3.02i)T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - 2.32iT - 3T^{2} \) |
| 5 | \( 1 - 4.13iT - 5T^{2} \) |
| 7 | \( 1 - 2.86iT - 7T^{2} \) |
| 11 | \( 1 + 0.105iT - 11T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 19 | \( 1 - 6.25T + 19T^{2} \) |
| 23 | \( 1 - 3.72iT - 23T^{2} \) |
| 29 | \( 1 - 0.0883iT - 29T^{2} \) |
| 31 | \( 1 + 5.14iT - 31T^{2} \) |
| 37 | \( 1 + 7.58iT - 37T^{2} \) |
| 41 | \( 1 - 9.44iT - 41T^{2} \) |
| 43 | \( 1 + 1.19T + 43T^{2} \) |
| 47 | \( 1 + 4.55T + 47T^{2} \) |
| 53 | \( 1 + 4.31T + 53T^{2} \) |
| 61 | \( 1 - 1.14iT - 61T^{2} \) |
| 67 | \( 1 - 5.82T + 67T^{2} \) |
| 71 | \( 1 - 13.0iT - 71T^{2} \) |
| 73 | \( 1 + 14.1iT - 73T^{2} \) |
| 79 | \( 1 + 14.0iT - 79T^{2} \) |
| 83 | \( 1 - 5.02T + 83T^{2} \) |
| 89 | \( 1 - 8.02T + 89T^{2} \) |
| 97 | \( 1 - 18.2iT - 97T^{2} \) |
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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
−9.237717178495505335614418988610, −8.216375268320177481096786146268, −7.48608858521870124451274550636, −6.47335543992685132602694359088, −5.96770022891022631112232473522, −5.29524488936068518956347288842, −4.12296325812766523857139473872, −3.45732157802142558932560679591, −2.95643614861324328073972359796, −1.88263009298966631049669139509,
0.62693400868630221952067565542, 1.08893457103619057267010367140, 1.86922202743625425087377611175, 3.32311413287683852314471719476, 4.29629767025786593868429021599, 4.97715287391646065987972252576, 5.79094614304027581400429236968, 6.70878745144739546355053014307, 7.22692092689556295378890640351, 8.115486743778326634324759646810