| L(s) = 1 | + (0.415 + 0.909i)2-s + (−0.326 − 0.209i)3-s + (−0.654 + 0.755i)4-s + (0.142 + 0.989i)5-s + (0.0552 − 0.384i)6-s + (1.81 + 3.97i)7-s + (−0.959 − 0.281i)8-s + (−1.18 − 2.59i)9-s + (−0.841 + 0.540i)10-s + (0.405 + 2.82i)11-s + (0.372 − 0.109i)12-s + (0.691 − 0.203i)13-s + (−2.86 + 3.30i)14-s + (0.161 − 0.353i)15-s + (−0.142 − 0.989i)16-s + (−2.18 − 2.52i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (−0.188 − 0.121i)3-s + (−0.327 + 0.377i)4-s + (0.0636 + 0.442i)5-s + (0.0225 − 0.156i)6-s + (0.686 + 1.50i)7-s + (−0.339 − 0.0996i)8-s + (−0.394 − 0.863i)9-s + (−0.266 + 0.170i)10-s + (0.122 + 0.850i)11-s + (0.107 − 0.0315i)12-s + (0.191 − 0.0563i)13-s + (−0.765 + 0.883i)14-s + (0.0416 − 0.0911i)15-s + (−0.0355 − 0.247i)16-s + (−0.530 − 0.612i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.497229 + 1.33045i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.497229 + 1.33045i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 67 | \( 1 + (-8.15 + 0.718i)T \) |
| good | 3 | \( 1 + (0.326 + 0.209i)T + (1.24 + 2.72i)T^{2} \) |
| 7 | \( 1 + (-1.81 - 3.97i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.405 - 2.82i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.691 + 0.203i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (2.18 + 2.52i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (2.11 - 4.62i)T + (-12.4 - 14.3i)T^{2} \) |
| 23 | \( 1 + (-3.54 - 2.28i)T + (9.55 + 20.9i)T^{2} \) |
| 29 | \( 1 + 9.08T + 29T^{2} \) |
| 31 | \( 1 + (-1.05 - 0.309i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 41 | \( 1 + (-6.94 - 8.01i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.267 - 0.309i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (7.25 + 4.65i)T + (19.5 + 42.7i)T^{2} \) |
| 53 | \( 1 + (2.91 - 3.35i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-3.59 - 1.05i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (0.536 - 3.72i)T + (-58.5 - 17.1i)T^{2} \) |
| 71 | \( 1 + (-2.75 + 3.17i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (0.0908 - 0.632i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-8.21 + 2.41i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.18 + 8.22i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-11.5 + 7.42i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−11.16049652283463933555593177134, −9.646330323217409779091347259498, −9.088159870668758084197079550386, −8.175195952771155815092766729411, −7.23406278339627231140215500130, −6.22985568805000013057292960292, −5.63562622743524668992389332354, −4.62291561223311477014696065497, −3.29232154425287558605092544632, −2.01984081357920483150246280901,
0.71142015533427525529675857541, 2.14043722529764097668516010116, 3.70051584354355267956984494156, 4.53134996594956314312470481833, 5.31674327437779953558183923895, 6.51157441901177055304280303940, 7.68881580025409241045932614158, 8.496421482280001597836872585476, 9.395950035410509891924197896267, 10.67739599002232692029770314712
