L(s) = 1 | + 0.0950i·3-s − 3.14i·7-s + 2.99·9-s − 1.67·11-s + 6.63i·13-s − 5.16i·17-s + 4.72·19-s + 0.299·21-s − 8.82i·23-s + 0.569i·27-s + 1.80·29-s − 1.65·31-s − 0.159i·33-s + 1.99i·37-s − 0.630·39-s + ⋯ |
L(s) = 1 | + 0.0549i·3-s − 1.18i·7-s + 0.996·9-s − 0.505·11-s + 1.83i·13-s − 1.25i·17-s + 1.08·19-s + 0.0652·21-s − 1.83i·23-s + 0.109i·27-s + 0.336·29-s − 0.298·31-s − 0.0277i·33-s + 0.327i·37-s − 0.100·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.918330543\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.918330543\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 - 0.0950iT - 3T^{2} \) |
| 7 | \( 1 + 3.14iT - 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 - 6.63iT - 13T^{2} \) |
| 17 | \( 1 + 5.16iT - 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 + 8.82iT - 23T^{2} \) |
| 29 | \( 1 - 1.80T + 29T^{2} \) |
| 31 | \( 1 + 1.65T + 31T^{2} \) |
| 37 | \( 1 - 1.99iT - 37T^{2} \) |
| 43 | \( 1 - 1.46iT - 43T^{2} \) |
| 47 | \( 1 - 8.53iT - 47T^{2} \) |
| 53 | \( 1 + 9.35iT - 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 9.67iT - 67T^{2} \) |
| 71 | \( 1 + 0.776T + 71T^{2} \) |
| 73 | \( 1 - 8.33iT - 73T^{2} \) |
| 79 | \( 1 + 0.915T + 79T^{2} \) |
| 83 | \( 1 + 10.0iT - 83T^{2} \) |
| 89 | \( 1 - 6.44T + 89T^{2} \) |
| 97 | \( 1 - 8.32iT - 97T^{2} \) |
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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
−8.195670319961422113587398497009, −7.36067209846619659003757840120, −6.95647611652620490649737993810, −6.37920741521496932270477686977, −4.98117617033124737147901052651, −4.55339065830231545331046780650, −3.86772999576708696731385667230, −2.78493096707027308487155970811, −1.68054963858447922746781576200, −0.62605776915274697799387293831,
1.10682322114801809465555024742, 2.14316691640828768757521265591, 3.14390690256427304865427339336, 3.80515991574207519714244434373, 5.13294528643273390318889038436, 5.50127480233327155227430483408, 6.15749101847444332824207168628, 7.35189384823055744855410013296, 7.74727413279815474833980505477, 8.493856915771222576753194924678