| L(s) = 1 | + (1.46 + 0.154i)2-s + (0.0450 − 0.211i)3-s + (0.179 + 0.0380i)4-s + (0.796 − 2.08i)5-s + (0.0988 − 0.304i)6-s + (2.55 − 0.703i)7-s + (−2.55 − 0.829i)8-s + (2.69 + 1.20i)9-s + (1.49 − 2.94i)10-s + (−5.12 + 2.28i)11-s + (0.0161 − 0.0362i)12-s + (2.43 + 3.35i)13-s + (3.85 − 0.639i)14-s + (−0.406 − 0.262i)15-s + (−3.95 − 1.76i)16-s + (−1.54 + 1.39i)17-s + ⋯ |
| L(s) = 1 | + (1.03 + 0.109i)2-s + (0.0259 − 0.122i)3-s + (0.0895 + 0.0190i)4-s + (0.356 − 0.934i)5-s + (0.0403 − 0.124i)6-s + (0.964 − 0.265i)7-s + (−0.902 − 0.293i)8-s + (0.899 + 0.400i)9-s + (0.472 − 0.931i)10-s + (−1.54 + 0.688i)11-s + (0.00465 − 0.0104i)12-s + (0.675 + 0.930i)13-s + (1.03 − 0.170i)14-s + (−0.104 − 0.0678i)15-s + (−0.989 − 0.440i)16-s + (−0.374 + 0.337i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.84181 - 0.311246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84181 - 0.311246i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.796 + 2.08i)T \) |
| 7 | \( 1 + (-2.55 + 0.703i)T \) |
| good | 2 | \( 1 + (-1.46 - 0.154i)T + (1.95 + 0.415i)T^{2} \) |
| 3 | \( 1 + (-0.0450 + 0.211i)T + (-2.74 - 1.22i)T^{2} \) |
| 11 | \( 1 + (5.12 - 2.28i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-2.43 - 3.35i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.54 - 1.39i)T + (1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.18 + 0.252i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (5.39 + 0.566i)T + (22.4 + 4.78i)T^{2} \) |
| 29 | \( 1 + (-0.466 - 1.43i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.46 + 1.62i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (0.927 - 2.08i)T + (-24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (-0.645 + 0.469i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 4.15iT - 43T^{2} \) |
| 47 | \( 1 + (-9.01 - 8.11i)T + (4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (2.22 - 10.4i)T + (-48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (1.05 + 10.0i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-0.761 + 7.24i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (1.81 - 1.62i)T + (7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (2.88 + 8.89i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.425 - 0.955i)T + (-48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (0.952 - 1.05i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (12.1 + 3.95i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.530 + 5.05i)T + (-87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (-9.10 + 2.95i)T + (78.4 - 57.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.84637018974957201419095327941, −12.13579357804873871921169888033, −10.78251842720514977282692713355, −9.682254371004640349602808775574, −8.491773715652314525115176568213, −7.43886872526964030541520117885, −5.89381659310628523664413139653, −4.79801515427603529669423740555, −4.27296988419075348675126548773, −1.93707692450164127467467897177,
2.57883729441445974331206511851, 3.80909719574531776167978905372, 5.21963432892361709048347684910, 5.98525732537049280248206142225, 7.53013993528848803080082403315, 8.603714831638042357210084583163, 10.08873900246365715584378899163, 10.89283500257660983214816032054, 11.88449932096167314745198679082, 13.07340617183331965703120286554
