| L(s) = 1 | + 1.18i·3-s + 0.609i·5-s + 4.31i·7-s + 1.58·9-s − 5.18i·11-s − 0.0691·13-s − 0.725·15-s + (−3.60 − 1.99i)17-s + 0.848·19-s − 5.12·21-s + 4.06i·23-s + 4.62·25-s + 5.45i·27-s + 4.23i·29-s − 2.80i·31-s + ⋯ |
| L(s) = 1 | + 0.686i·3-s + 0.272i·5-s + 1.62i·7-s + 0.528·9-s − 1.56i·11-s − 0.0191·13-s − 0.187·15-s + (−0.874 − 0.484i)17-s + 0.194·19-s − 1.11·21-s + 0.848i·23-s + 0.925·25-s + 1.04i·27-s + 0.787i·29-s − 0.503i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.505526449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.505526449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (3.60 + 1.99i)T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 1.18iT - 3T^{2} \) |
| 5 | \( 1 - 0.609iT - 5T^{2} \) |
| 7 | \( 1 - 4.31iT - 7T^{2} \) |
| 11 | \( 1 + 5.18iT - 11T^{2} \) |
| 13 | \( 1 + 0.0691T + 13T^{2} \) |
| 19 | \( 1 - 0.848T + 19T^{2} \) |
| 23 | \( 1 - 4.06iT - 23T^{2} \) |
| 29 | \( 1 - 4.23iT - 29T^{2} \) |
| 31 | \( 1 + 2.80iT - 31T^{2} \) |
| 37 | \( 1 - 8.93iT - 37T^{2} \) |
| 41 | \( 1 - 0.703iT - 41T^{2} \) |
| 43 | \( 1 + 0.966T + 43T^{2} \) |
| 47 | \( 1 + 0.386T + 47T^{2} \) |
| 53 | \( 1 + 0.833T + 53T^{2} \) |
| 61 | \( 1 - 12.4iT - 61T^{2} \) |
| 67 | \( 1 + 5.54T + 67T^{2} \) |
| 71 | \( 1 - 3.43iT - 71T^{2} \) |
| 73 | \( 1 - 11.9iT - 73T^{2} \) |
| 79 | \( 1 - 11.6iT - 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 3.15T + 89T^{2} \) |
| 97 | \( 1 - 9.31iT - 97T^{2} \) |
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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
−8.743754443673466196887064475198, −8.370006317800394957448879923672, −7.20717295985108318071250383578, −6.46326121000554473274229584040, −5.63578071792801912544442233331, −5.15932267784240787118059024076, −4.21674572612776158453708165226, −3.15800510930278963613781466425, −2.71317164662775036028195318226, −1.36714968482711899009781208617,
0.44170510546511912602839874341, 1.50004803425938915353821300824, 2.24919997258119633806263792837, 3.66821005116947694145788660983, 4.50362832654870405174479287923, 4.74899644208815751039177569939, 6.27094968703031504190810171593, 6.82131911872453754777056649108, 7.39902403520009545181550900476, 7.80828499907456186610585593134
