L(s) = 1 | + (−0.189 − 0.328i)3-s + (−0.5 − 0.866i)5-s − 1.89·7-s + (1.42 − 2.47i)9-s − 0.134·11-s + (−1.75 + 3.04i)13-s + (−0.189 + 0.328i)15-s + (0.830 + 1.43i)17-s + (−2.10 − 3.81i)19-s + (0.359 + 0.621i)21-s + (2.68 − 4.65i)23-s + (−0.499 + 0.866i)25-s − 2.22·27-s + (−2.48 + 4.30i)29-s − 6.56·31-s + ⋯ |
L(s) = 1 | + (−0.109 − 0.189i)3-s + (−0.223 − 0.387i)5-s − 0.715·7-s + (0.476 − 0.824i)9-s − 0.0405·11-s + (−0.487 + 0.843i)13-s + (−0.0489 + 0.0848i)15-s + (0.201 + 0.348i)17-s + (−0.483 − 0.875i)19-s + (0.0783 + 0.135i)21-s + (0.559 − 0.969i)23-s + (−0.0999 + 0.173i)25-s − 0.427·27-s + (−0.461 + 0.799i)29-s − 1.17·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2051606135\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2051606135\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (2.10 + 3.81i)T \) |
good | 3 | \( 1 + (0.189 + 0.328i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 1.89T + 7T^{2} \) |
| 11 | \( 1 + 0.134T + 11T^{2} \) |
| 13 | \( 1 + (1.75 - 3.04i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.830 - 1.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.68 + 4.65i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.48 - 4.30i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 + 1.69T + 37T^{2} \) |
| 41 | \( 1 + (5.31 + 9.20i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.25 - 7.36i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.55 - 9.62i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.132 + 0.229i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.44 + 5.97i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.58 - 7.94i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.47 - 2.55i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.664 - 1.15i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.17 - 5.49i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.733 + 1.27i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 7.44T + 83T^{2} \) |
| 89 | \( 1 + (4.86 - 8.43i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.73 + 15.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.218533502601287286235935209803, −8.355209226063966526157640088860, −7.12128345077768583163870633923, −6.81265250490521818448562821953, −5.84845944077527642992754718125, −4.74306034816901942716963773820, −3.95791297018816450298980362253, −2.92881321296041894219250331758, −1.55082738815426603998228757966, −0.079465659315318079634297236963,
1.84562970725537258309089945938, 3.06911508356420823020047129651, 3.85100646275465312001260682401, 5.01855245397624295395470435755, 5.71335706095580309807124101733, 6.75577924479489807625790301007, 7.54661847795971302891303492943, 8.093333469958397038730056392703, 9.295004206791034694842500681008, 9.982061422963699771565645924511