L(s) = 1 | + (−0.520 − 1.60i)2-s + (−0.574 + 0.417i)3-s + (−0.674 + 0.490i)4-s + (0.967 + 0.702i)6-s + 4.59·7-s + (−1.58 − 1.15i)8-s + (−0.771 + 2.37i)9-s + (1.20 + 3.72i)11-s + (0.183 − 0.563i)12-s + (0.177 − 0.544i)13-s + (−2.38 − 7.35i)14-s + (−1.53 + 4.72i)16-s + (−0.188 − 0.136i)17-s + 4.20·18-s + (4.49 + 3.26i)19-s + ⋯ |
L(s) = 1 | + (−0.367 − 1.13i)2-s + (−0.331 + 0.241i)3-s + (−0.337 + 0.245i)4-s + (0.394 + 0.286i)6-s + 1.73·7-s + (−0.561 − 0.407i)8-s + (−0.257 + 0.791i)9-s + (0.364 + 1.12i)11-s + (0.0528 − 0.162i)12-s + (0.0491 − 0.151i)13-s + (−0.638 − 1.96i)14-s + (−0.384 + 1.18i)16-s + (−0.0456 − 0.0331i)17-s + 0.990·18-s + (1.03 + 0.748i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25430 - 0.436794i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25430 - 0.436794i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (0.520 + 1.60i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (0.574 - 0.417i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 - 4.59T + 7T^{2} \) |
| 11 | \( 1 + (-1.20 - 3.72i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.177 + 0.544i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.188 + 0.136i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-4.49 - 3.26i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.52 - 4.69i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (3.34 - 2.42i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.82 - 2.05i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.67 + 5.15i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.22 + 9.91i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 1.38T + 43T^{2} \) |
| 47 | \( 1 + (-0.744 + 0.541i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.996 + 0.723i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.39 + 4.28i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (3.60 + 11.0i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (2.39 + 1.73i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (2.59 - 1.88i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.18 - 9.78i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.77 + 5.65i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (8.48 + 6.16i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.24 - 6.90i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (6.73 - 4.89i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.71881597278495686423832317953, −9.892364199288304975252489388861, −9.043806572213234736465700666278, −7.974937358734034860980228095780, −7.23538652720830393633859148888, −5.63191866223625442580576355765, −4.89572877813278269685527832744, −3.78021846593826299893173429307, −2.21494895577632750567558289083, −1.43275591428723101185246676295,
1.00846418000280437145098707160, 2.88427432415601900953818462106, 4.51495959017902862770571963011, 5.54542599649787979044179842396, 6.25869192365132769530768797343, 7.17841513189982924628644715834, 8.060465894368585507991087068010, 8.640655313196870187152775248387, 9.426067816947944608169232891099, 10.97930713932371707743336612226