| L(s) = 1 | + 0.902i·3-s + 0.891i·7-s + 2.18·9-s − 0.804·21-s + 3.04i·23-s + 4.67i·27-s + 0.403·29-s + 3.43·41-s − 1.13i·43-s + 7.82i·47-s + 6.20·49-s − 3.46·61-s + 1.94i·63-s + 14.1i·67-s − 2.74·69-s + ⋯ |
| L(s) = 1 | + 0.520i·3-s + 0.336i·7-s + 0.728·9-s − 0.175·21-s + 0.635i·23-s + 0.900i·27-s + 0.0748·29-s + 0.536·41-s − 0.172i·43-s + 1.14i·47-s + 0.886·49-s − 0.443·61-s + 0.245i·63-s + 1.73i·67-s − 0.330·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.861970552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.861970552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.902iT - 3T^{2} \) |
| 7 | \( 1 - 0.891iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 3.04iT - 23T^{2} \) |
| 29 | \( 1 - 0.403T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 + 1.13iT - 43T^{2} \) |
| 47 | \( 1 - 7.82iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 - 14.1iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 15.5iT - 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.852758292964859763322004803591, −7.80113892279838245315834132499, −7.28556713563846482876308679909, −6.37141466160232890514301735484, −5.61231085621493620541464017919, −4.81119127184203426336565773232, −4.10636613340677726041354376455, −3.29539306889793677538660948188, −2.26009220005354461782973869696, −1.14849613316856553100799179305,
0.59680998547117400906819491012, 1.65832261930083873131831508860, 2.58780830653611020963398655993, 3.71170451602661189269095506454, 4.42663380413081121310575497751, 5.26628954529932169917760323345, 6.26057996463337849094535932507, 6.81877938092145426542807913199, 7.52403535103178599802949645954, 8.108231474387351440122375336780
