| L(s) = 1 | + (−0.255 − 0.283i)2-s + (0.704 − 1.58i)3-s + (0.193 − 1.84i)4-s + (−0.827 + 0.918i)5-s + (−0.629 + 0.204i)6-s + (−0.913 + 0.406i)7-s + (−1.19 + 0.865i)8-s + (−2.00 − 2.22i)9-s + 0.472·10-s + (2.70 + 1.91i)11-s + (−2.78 − 1.60i)12-s + (6.33 − 1.34i)13-s + (0.348 + 0.155i)14-s + (0.870 + 1.95i)15-s + (−3.07 − 0.654i)16-s + (1.5 + 4.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.180 − 0.200i)2-s + (0.406 − 0.913i)3-s + (0.0969 − 0.921i)4-s + (−0.369 + 0.410i)5-s + (−0.256 + 0.0834i)6-s + (−0.345 + 0.153i)7-s + (−0.421 + 0.305i)8-s + (−0.669 − 0.743i)9-s + 0.149·10-s + (0.815 + 0.578i)11-s + (−0.802 − 0.463i)12-s + (1.75 − 0.373i)13-s + (0.0932 + 0.0415i)14-s + (0.224 + 0.504i)15-s + (−0.769 − 0.163i)16-s + (0.363 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.768944 - 0.630767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.768944 - 0.630767i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.704 + 1.58i)T \) |
| 11 | \( 1 + (-2.70 - 1.91i)T \) |
| good | 2 | \( 1 + (0.255 + 0.283i)T + (-0.209 + 1.98i)T^{2} \) |
| 5 | \( 1 + (0.827 - 0.918i)T + (-0.522 - 4.97i)T^{2} \) |
| 7 | \( 1 + (0.913 - 0.406i)T + (4.68 - 5.20i)T^{2} \) |
| 13 | \( 1 + (-6.33 + 1.34i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 4.61i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (2.30 + 3.99i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.43 - 1.97i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (0.604 - 0.128i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-4.11 - 2.99i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (2.30 + 1.02i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.927 - 1.60i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.24 + 11.8i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-1.26 + 3.88i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (0.169 - 1.60i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (10.6 + 2.25i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (3 + 5.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.899 + 2.76i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.118 - 0.0857i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-7.06 - 7.84i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-9.55 - 2.03i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + 6.76T + 89T^{2} \) |
| 97 | \( 1 + (-4.01 - 4.45i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−13.68811344843099023361237611772, −12.65445249087372639557798060099, −11.51870867076542563255055400345, −10.56643001867276432102484798869, −9.237493070257566844610889167366, −8.214112940045769220157459935219, −6.69300762539668280481065120313, −5.96203555161875450458734773773, −3.56882269449676017353387022836, −1.60727503260737922382578774019,
3.33698196508211097243001960173, 4.20861360920313053861576519602, 6.14866934119995292180428372767, 7.75426052040894840684884662946, 8.754612452704101340198826113447, 9.436433980401873231852316417704, 11.09945145322363421816600052182, 11.81817531259072348253916400798, 13.27850289653267878154040147468, 14.04326566683402629408857342944
