L(s) = 1 | + (−1.5 − 2.59i)3-s + (0.5 − 0.866i)5-s + (−0.5 + 2.59i)7-s + (−3 + 5.19i)9-s + (−1 − 1.73i)11-s − 6·13-s − 3·15-s + (−1 − 1.73i)17-s + (7.5 − 2.59i)21-s + (−4.5 + 7.79i)23-s + (−0.499 − 0.866i)25-s + 9·27-s + 3·29-s + (1 + 1.73i)31-s + (−3 + 5.19i)33-s + ⋯ |
L(s) = 1 | + (−0.866 − 1.49i)3-s + (0.223 − 0.387i)5-s + (−0.188 + 0.981i)7-s + (−1 + 1.73i)9-s + (−0.301 − 0.522i)11-s − 1.66·13-s − 0.774·15-s + (−0.242 − 0.420i)17-s + (1.63 − 0.566i)21-s + (−0.938 + 1.62i)23-s + (−0.0999 − 0.173i)25-s + 1.73·27-s + 0.557·29-s + (0.179 + 0.311i)31-s + (−0.522 + 0.904i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 3 | \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (7 + 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - T + 83T^{2} \) |
| 89 | \( 1 + (6.5 - 11.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.19928638796643000909459275249, −9.229950720918807290995336527717, −8.141731613205317063466886155332, −7.40348490250964578414996655636, −6.45656769547870055658996455581, −5.59833468637058787611191790817, −4.98578943868135646803071325141, −2.78382669842538044614324828656, −1.73440781798954448186915495075, 0,
2.67102895762323323698297291756, 4.17734235668272847511171272854, 4.58956822920757238304778436227, 5.73764375565985845850705092204, 6.71468216381318829784125468996, 7.69300552879846633339242529603, 9.135668869186383838277426400648, 10.03491122470901708091516793093, 10.32405943704433599832285826342