L(s) = 1 | + (−0.764 − 2.35i)2-s + (1.71 − 1.24i)3-s + (−3.33 + 2.42i)4-s + (−4.23 − 3.07i)6-s + 0.973·7-s + (4.24 + 3.08i)8-s + (0.457 − 1.40i)9-s + (−1.66 − 5.11i)11-s + (−2.69 + 8.29i)12-s + (0.617 − 1.89i)13-s + (−0.744 − 2.28i)14-s + (1.46 − 4.51i)16-s + (−1.65 − 1.20i)17-s − 3.66·18-s + (−5.01 − 3.64i)19-s + ⋯ |
L(s) = 1 | + (−0.540 − 1.66i)2-s + (0.988 − 0.718i)3-s + (−1.66 + 1.21i)4-s + (−1.72 − 1.25i)6-s + 0.367·7-s + (1.50 + 1.09i)8-s + (0.152 − 0.469i)9-s + (−0.501 − 1.54i)11-s + (−0.778 + 2.39i)12-s + (0.171 − 0.526i)13-s + (−0.198 − 0.611i)14-s + (0.366 − 1.12i)16-s + (−0.400 − 0.291i)17-s − 0.863·18-s + (−1.15 − 0.836i)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
\(L(1)\) |
\(\approx\) |
\(0.389142 + 1.11746i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.389142 + 1.11746i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.764 + 2.35i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.71 + 1.24i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 - 0.973T + 7T^{2} \) |
| 11 | \( 1 + (1.66 + 5.11i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.617 + 1.89i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (1.65 + 1.20i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (5.01 + 3.64i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.598 + 1.84i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (3.89 - 2.83i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-5.37 - 3.90i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.302 + 0.931i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.844 + 2.59i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 3.99T + 43T^{2} \) |
| 47 | \( 1 + (-5.83 + 4.23i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-10.7 + 7.80i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.02 + 6.22i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.842 - 2.59i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-7.74 - 5.62i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-4.59 + 3.34i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.89 - 8.89i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.57 + 1.87i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.94 - 3.59i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.929 + 2.86i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.81 + 3.50i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.31393779913521583505863888895, −9.051975824175865044307485090432, −8.484308341885002598492289867198, −8.104526082674285959047016176234, −6.75923345472278355916669498629, −5.17955419469011646838268306465, −3.75970148418105207402106544327, −2.84622465820546800688201692505, −2.11676736993369166061875678156, −0.68700345732693732141276609237,
2.17501783487276168944968497319, 4.09461354445733826254986421776, 4.65925235654210628731130502331, 5.92355790546924485483624881534, 6.85678440729393946869941084232, 7.86181607126524440699395835425, 8.297039538443093108887611786384, 9.270488156735553639325951746981, 9.754788605601218114532072189961, 10.56983160634826285187811623943