| L(s) = 1 | + (−0.261 + 1.38i)2-s + (1.60 + 0.661i)3-s + (−1.86 − 0.728i)4-s + (−1.33 + 2.05i)6-s + (1.49 − 2.39i)8-s + (2.12 + 2.11i)9-s + (−2.5 − 2.39i)12-s + 6.88i·13-s + (2.93 + 2.71i)16-s + (−3.5 + 2.39i)18-s + 4.79i·23-s + (3.98 − 2.84i)24-s + 5·25-s + (−9.56 − 1.80i)26-s + (2.00 + 4.79i)27-s + ⋯ |
| L(s) = 1 | + (−0.185 + 0.982i)2-s + (0.924 + 0.381i)3-s + (−0.931 − 0.364i)4-s + (−0.546 + 0.837i)6-s + (0.530 − 0.847i)8-s + (0.708 + 0.705i)9-s + (−0.721 − 0.692i)12-s + 1.90i·13-s + (0.734 + 0.678i)16-s + (−0.824 + 0.565i)18-s + 0.999i·23-s + (0.813 − 0.580i)24-s + 25-s + (−1.87 − 0.353i)26-s + (0.384 + 0.922i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.733303 + 1.42434i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.733303 + 1.42434i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.261 - 1.38i)T \) |
| 3 | \( 1 + (-1.60 - 0.661i)T \) |
| 23 | \( 1 - 4.79iT \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 6.88iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 9.52iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 7.14iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 15.0iT - 71T^{2} \) |
| 73 | \( 1 + 17.0T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−10.82217127902789289959053541286, −9.796511130462047858336367344003, −9.119925280099906259581833835979, −8.597850529585521486193438860224, −7.40763200415435186506245973604, −6.90167769571517369360690049203, −5.56086905465484851862354467684, −4.46616212673430863925703752106, −3.66524970099087195330454584975, −1.86404629809685411810563817954,
0.971435110153594664429244144556, 2.52743280242562549898377792052, 3.26696809802342000724666918599, 4.44390053800345717176284282676, 5.72327801134497485670418218179, 7.22875783976663589950859438060, 8.116078014204282121010141215146, 8.679208753167622524999275941300, 9.718275129264906517310728677582, 10.36615356248421411590846422050
