| L(s) = 1 | + (−1.12 − 0.861i)2-s + (0.416 + 0.0828i)3-s + (0.514 + 1.93i)4-s + (1.82 − 2.73i)5-s + (−0.395 − 0.451i)6-s + (0.00395 − 0.00163i)7-s + (1.08 − 2.61i)8-s + (−2.60 − 1.07i)9-s + (−4.39 + 1.48i)10-s + (0.744 + 3.74i)11-s + (0.0542 + 0.847i)12-s + (2.10 + 3.15i)13-s + (−0.00583 − 0.00156i)14-s + (0.985 − 0.985i)15-s + (−3.47 + 1.98i)16-s + (−0.937 − 0.937i)17-s + ⋯ |
| L(s) = 1 | + (−0.792 − 0.609i)2-s + (0.240 + 0.0478i)3-s + (0.257 + 0.966i)4-s + (0.815 − 1.22i)5-s + (−0.161 − 0.184i)6-s + (0.00149 − 0.000618i)7-s + (0.384 − 0.922i)8-s + (−0.868 − 0.359i)9-s + (−1.39 + 0.470i)10-s + (0.224 + 1.12i)11-s + (0.0156 + 0.244i)12-s + (0.584 + 0.874i)13-s + (−0.00156 − 0.000419i)14-s + (0.254 − 0.254i)15-s + (−0.867 + 0.497i)16-s + (−0.227 − 0.227i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.655123 - 0.311702i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.655123 - 0.311702i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.12 + 0.861i)T \) |
| good | 3 | \( 1 + (-0.416 - 0.0828i)T + (2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (-1.82 + 2.73i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-0.00395 + 0.00163i)T + (4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.744 - 3.74i)T + (-10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-2.10 - 3.15i)T + (-4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (0.937 + 0.937i)T + 17iT^{2} \) |
| 19 | \( 1 + (6.36 - 4.25i)T + (7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (-0.0154 + 0.0374i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-0.237 + 1.19i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + 5.05iT - 31T^{2} \) |
| 37 | \( 1 + (-5.65 - 3.78i)T + (14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (1.75 - 4.24i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-10.6 + 2.12i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (7.35 + 7.35i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.15 + 5.82i)T + (-48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (-3.18 + 4.76i)T + (-22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (-2.17 - 0.431i)T + (56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (2.42 + 0.483i)T + (61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (10.9 - 4.52i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-4.86 - 2.01i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-10.3 + 10.3i)T - 79iT^{2} \) |
| 83 | \( 1 + (-0.0153 + 0.0102i)T + (31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (3.25 + 7.86i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 8.60iT - 97T^{2} \) |
| 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
−14.75147314821932228620197459702, −13.36139408256112359543163942276, −12.49322280863912448011440913155, −11.44019566832738814245312934218, −9.874670992437338795971625623196, −9.111188638835385114780569495381, −8.218427798637017188474598000429, −6.31629049197268851966878529428, −4.29777971477897202438219844687, −1.94837166900738108115012747540,
2.69710331255540056783903452256, 5.75245556120467161687054988524, 6.57777623409680311660623648658, 8.143550281459692432738737016444, 9.127961665850475215351191231410, 10.78779831422323111518957670828, 10.91448185561292879504455886969, 13.37502824624126273558981280639, 14.26491720049914768296433555166, 14.99238786712003394446548766473
