L(s) = 1 | + (1.67 + 0.448i)2-s + (1.73 + 1.00i)4-s + (1.22 + 1.22i)8-s + (−0.866 + 0.5i)9-s + (0.5 − 0.866i)11-s + (0.500 + 0.866i)16-s + (−1.67 + 0.448i)18-s + (1.22 − 1.22i)22-s + (−0.448 + 1.67i)23-s − i·29-s − 2·36-s + (−1.67 − 0.448i)37-s + (−1.22 − 1.22i)43-s + (1.73 − 0.999i)44-s + (−1.50 + 2.59i)46-s + ⋯ |
L(s) = 1 | + (1.67 + 0.448i)2-s + (1.73 + 1.00i)4-s + (1.22 + 1.22i)8-s + (−0.866 + 0.5i)9-s + (0.5 − 0.866i)11-s + (0.500 + 0.866i)16-s + (−1.67 + 0.448i)18-s + (1.22 − 1.22i)22-s + (−0.448 + 1.67i)23-s − i·29-s − 2·36-s + (−1.67 − 0.448i)37-s + (−1.22 − 1.22i)43-s + (1.73 − 0.999i)44-s + (−1.50 + 2.59i)46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.525007766\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.525007766\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 3 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−10.20975110728442040173566623977, −9.013240639760521944739305929007, −8.149833043429357400684200940703, −7.29786471094650255297087757286, −6.39058873451255708450347278103, −5.61893870590865796451789136215, −5.14345942101640519673494507637, −3.86968123972594648253967970894, −3.31354517513423685009325684271, −2.11991578274121113102865490538,
1.80517201745387408000729403296, 2.89298250954082879827283951574, 3.71606341840407907275218434847, 4.67131232543198701122677113204, 5.33283768543182547513374423079, 6.47308833896276499174317501134, 6.73398714500897558099284955613, 8.174181667813337603160175118819, 9.089357583040083784776658982188, 10.13308979483274934074866268652