| L(s) = 1 | + (−1.36 − 0.366i)2-s + (2.82 − 1.00i)3-s + (1.73 + i)4-s + (2.52 + 4.31i)5-s + (−4.22 + 0.344i)6-s + (0.692 − 6.96i)7-s + (−1.99 − 2i)8-s + (6.96 − 5.70i)9-s + (−1.86 − 6.82i)10-s + (3.24 + 1.87i)11-s + (5.90 + 1.07i)12-s + (5.63 + 5.63i)13-s + (−3.49 + 9.26i)14-s + (11.4 + 9.64i)15-s + (1.99 + 3.46i)16-s + (−4.24 + 1.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.683 − 0.183i)2-s + (0.941 − 0.336i)3-s + (0.433 + 0.250i)4-s + (0.504 + 0.863i)5-s + (−0.704 + 0.0574i)6-s + (0.0989 − 0.995i)7-s + (−0.249 − 0.250i)8-s + (0.773 − 0.633i)9-s + (−0.186 − 0.682i)10-s + (0.294 + 0.170i)11-s + (0.491 + 0.0897i)12-s + (0.433 + 0.433i)13-s + (−0.249 + 0.661i)14-s + (0.765 + 0.643i)15-s + (0.124 + 0.216i)16-s + (−0.249 + 0.0669i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.71165 - 0.328611i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71165 - 0.328611i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 + 0.366i)T \) |
| 3 | \( 1 + (-2.82 + 1.00i)T \) |
| 5 | \( 1 + (-2.52 - 4.31i)T \) |
| 7 | \( 1 + (-0.692 + 6.96i)T \) |
| good | 11 | \( 1 + (-3.24 - 1.87i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-5.63 - 5.63i)T + 169iT^{2} \) |
| 17 | \( 1 + (4.24 - 1.13i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-8.60 - 14.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-9.56 + 35.7i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 - 29.5T + 841T^{2} \) |
| 31 | \( 1 + (11.0 + 6.36i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (6.31 - 23.5i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 26.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-38.5 + 38.5i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (7.31 - 27.3i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-7.09 + 1.90i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (68.7 + 39.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (47.6 - 27.5i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (79.7 - 21.3i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 86.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-9.35 + 2.50i)T + (4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (134. - 77.7i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-68.5 + 68.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (94.3 - 54.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (17.8 - 17.8i)T - 9.40e3iT^{2} \) |
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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
−12.02594993039150681051068654578, −10.75006780839887495670764606418, −10.13674978227592146513137017577, −9.149425583229310791159637418690, −8.106878905192551001300393919505, −7.09011484974156608151144575295, −6.42882611785335026743070251053, −4.10290745401407053359556903190, −2.85859954555855978115873247449, −1.44997919752649184300968620986,
1.56757027427735880026872910331, 3.01858152602025157125148250530, 4.84197804292192150687469218297, 5.93500108932754096870965606077, 7.48100911643962616143058301768, 8.588361020247157518123154992351, 9.073155916356161784720108950505, 9.801077139496100845658267556575, 11.05693160801134546110707487027, 12.22054600337259511362100683687
