| L(s) = 1 | + 2-s + 4-s + (−1.60 + 1.55i)5-s + 8-s + (−1.60 + 1.55i)10-s − 4.10i·11-s − 2.67·13-s + 16-s + 1.29i·17-s + 6.96i·19-s + (−1.60 + 1.55i)20-s − 4.10i·22-s − 3.53·23-s + (0.143 − 4.99i)25-s − 2.67·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.717 + 0.696i)5-s + 0.353·8-s + (−0.507 + 0.492i)10-s − 1.23i·11-s − 0.742·13-s + 0.250·16-s + 0.314i·17-s + 1.59i·19-s + (−0.358 + 0.348i)20-s − 0.875i·22-s − 0.736·23-s + (0.0287 − 0.999i)25-s − 0.525·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.138550518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.138550518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.60 - 1.55i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4.10iT - 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 - 1.29iT - 17T^{2} \) |
| 19 | \( 1 - 6.96iT - 19T^{2} \) |
| 23 | \( 1 + 3.53T + 23T^{2} \) |
| 29 | \( 1 - 3.22iT - 29T^{2} \) |
| 31 | \( 1 + 8.38iT - 31T^{2} \) |
| 37 | \( 1 + 11.3iT - 37T^{2} \) |
| 41 | \( 1 - 1.99T + 41T^{2} \) |
| 43 | \( 1 + 0.0984iT - 43T^{2} \) |
| 47 | \( 1 + 9.68iT - 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 0.796T + 59T^{2} \) |
| 61 | \( 1 - 2.69iT - 61T^{2} \) |
| 67 | \( 1 - 0.696iT - 67T^{2} \) |
| 71 | \( 1 + 9.32iT - 71T^{2} \) |
| 73 | \( 1 - 7.26T + 73T^{2} \) |
| 79 | \( 1 - 2.73T + 79T^{2} \) |
| 83 | \( 1 + 6.79iT - 83T^{2} \) |
| 89 | \( 1 + 8.03T + 89T^{2} \) |
| 97 | \( 1 - 6.29T + 97T^{2} \) |
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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
−7.82891724999514517375592115719, −7.56822224363258409295478116967, −6.48631225969512241326067931742, −5.96683978453643455216439508928, −5.26045883745821433051729623455, −4.04156987417547127043228109278, −3.73439873891765553320852226902, −2.83662552285672211387220676913, −1.89060263937599279265736608770, −0.24114300103729916006026037803,
1.27136282874333658399726317894, 2.42042101941384222191631030477, 3.23626411761024249215979118801, 4.42458343728792394859121137554, 4.67266776150863951100203440711, 5.30971897816736960902651924228, 6.52538385925356244520591018474, 7.05537587797887958867778964250, 7.76017792866459726821406623870, 8.404633453550558155231893515768
