L(s) = 1 | − 1.73i·2-s + (−0.5 − 0.866i)3-s − 0.999·4-s + (−1.5 + 0.866i)5-s + (−1.49 + 0.866i)6-s − 1.73i·8-s + (1 − 1.73i)9-s + (1.49 + 2.59i)10-s + (−4.5 + 2.59i)11-s + (0.499 + 0.866i)12-s + (1 − 3.46i)13-s + (1.5 + 0.866i)15-s − 5·16-s − 6·17-s + (−3 − 1.73i)18-s + (1.5 + 0.866i)19-s + ⋯ |
L(s) = 1 | − 1.22i·2-s + (−0.288 − 0.499i)3-s − 0.499·4-s + (−0.670 + 0.387i)5-s + (−0.612 + 0.353i)6-s − 0.612i·8-s + (0.333 − 0.577i)9-s + (0.474 + 0.821i)10-s + (−1.35 + 0.783i)11-s + (0.144 + 0.249i)12-s + (0.277 − 0.960i)13-s + (0.387 + 0.223i)15-s − 1.25·16-s − 1.45·17-s + (−0.707 − 0.408i)18-s + (0.344 + 0.198i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.267371 + 0.395611i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.267371 + 0.395611i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-1 + 3.46i)T \) |
good | 2 | \( 1 + 1.73iT - 2T^{2} \) |
| 3 | \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (7.5 - 4.33i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 3.46iT - 59T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.5 + 0.866i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + (-4.5 + 2.59i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.24681892052838759436974869280, −9.485150478624857185707624148994, −8.187769139952625408806874395660, −7.27285222135996009362554055102, −6.59635508802287782532496163457, −5.18780692382171607647907423357, −3.95933413418475588613507594495, −3.02418231469773764169373465398, −1.87852419102190085795320780970, −0.25148432323485417672786791210,
2.40279497122692555087689854232, 4.19914138268499238712459652604, 4.87230657305329020076787892974, 5.77607764717510825153228139047, 6.75416339491388564675262158490, 7.72100756015289648089441662630, 8.297475241199766100435902232246, 9.134862506305999947036118212250, 10.37087380128518449082953256803, 11.22783433584987940958054786872