L(s) = 1 | + (0.866 + 0.5i)3-s + (0.5 − 1.86i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 0.5i)13-s + (−1.36 − 0.366i)19-s + (1.36 − 1.36i)21-s + i·25-s + 0.999i·27-s + (−0.366 + 0.366i)31-s + (0.366 + 1.36i)37-s − 0.999·39-s + (0.5 + 0.866i)43-s + (−2.36 − 1.36i)49-s + (−0.999 − i)57-s + (−0.866 − 1.5i)61-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)3-s + (0.5 − 1.86i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 0.5i)13-s + (−1.36 − 0.366i)19-s + (1.36 − 1.36i)21-s + i·25-s + 0.999i·27-s + (−0.366 + 0.366i)31-s + (0.366 + 1.36i)37-s − 0.999·39-s + (0.5 + 0.866i)43-s + (−2.36 − 1.36i)49-s + (−0.999 − i)57-s + (−0.866 − 1.5i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.204667814\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.204667814\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.866 - 0.5i)T \) |
good | 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 37 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + (-0.366 + 0.366i)T - iT^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−10.69941595385522128318384696583, −9.985416981284775814364413091994, −9.177212072470621546888811377706, −8.104596583303451916414464294770, −7.44495704474233774776112083470, −6.65203453635774300029775452883, −4.81473958450904577445881581729, −4.33630652599484696112003490700, −3.26443349299923955959647998336, −1.75728726508139447476619980978,
2.10573740431087150174802393842, 2.65228169539454310578836083219, 4.18878779534206885630759197950, 5.46632740096823570222418438544, 6.29004420484763141937183977426, 7.50241911370540350028630008497, 8.337660594498790660203130587570, 8.879638004090138627527230885643, 9.698064287911136603158129280533, 10.80488735607858675521618633442