| L(s) = 1 | + (−0.876 + 1.84i)2-s + (2.05 + 1.89i)3-s + (−1.37 − 1.68i)4-s + (0.914 − 2.04i)5-s + (−5.30 + 2.13i)6-s + (1.58 + 0.459i)7-s + (0.359 − 0.0886i)8-s + (0.388 + 4.81i)9-s + (2.96 + 3.47i)10-s + (−1.42 − 1.68i)11-s + (0.368 − 6.09i)12-s + (−0.533 + 3.56i)13-s + (−2.24 + 2.52i)14-s + (5.75 − 2.45i)15-s + (0.721 − 3.53i)16-s + (3.76 + 6.83i)17-s + ⋯ |
| L(s) = 1 | + (−0.619 + 1.30i)2-s + (1.18 + 1.09i)3-s + (−0.689 − 0.844i)4-s + (0.409 − 0.912i)5-s + (−2.16 + 0.871i)6-s + (0.599 + 0.173i)7-s + (0.127 − 0.0313i)8-s + (0.129 + 1.60i)9-s + (0.938 + 1.10i)10-s + (−0.430 − 0.506i)11-s + (0.106 − 1.75i)12-s + (−0.147 + 0.988i)13-s + (−0.598 + 0.675i)14-s + (1.48 − 0.635i)15-s + (0.180 − 0.882i)16-s + (0.913 + 1.65i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.207232 + 1.74669i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.207232 + 1.74669i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.914 + 2.04i)T \) |
| 13 | \( 1 + (0.533 - 3.56i)T \) |
| good | 2 | \( 1 + (0.876 - 1.84i)T + (-1.26 - 1.54i)T^{2} \) |
| 3 | \( 1 + (-2.05 - 1.89i)T + (0.241 + 2.99i)T^{2} \) |
| 7 | \( 1 + (-1.58 - 0.459i)T + (5.91 + 3.74i)T^{2} \) |
| 11 | \( 1 + (1.42 + 1.68i)T + (-1.76 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-3.76 - 6.83i)T + (-9.08 + 14.3i)T^{2} \) |
| 19 | \( 1 + (-0.970 + 0.260i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (1.42 - 5.32i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (2.02 + 0.960i)T + (18.3 + 22.4i)T^{2} \) |
| 31 | \( 1 + (-0.00397 - 0.00311i)T + (7.41 + 30.0i)T^{2} \) |
| 37 | \( 1 + (0.591 + 0.787i)T + (-10.2 + 35.5i)T^{2} \) |
| 41 | \( 1 + (-1.76 - 1.62i)T + (3.29 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-0.999 - 7.04i)T + (-41.3 + 11.9i)T^{2} \) |
| 47 | \( 1 + (-0.689 + 0.261i)T + (35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (3.03 - 5.02i)T + (-24.6 - 46.9i)T^{2} \) |
| 59 | \( 1 + (0.790 + 0.522i)T + (23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (-9.22 + 9.60i)T + (-2.45 - 60.9i)T^{2} \) |
| 67 | \( 1 + (-3.19 + 3.91i)T + (-13.4 - 65.6i)T^{2} \) |
| 71 | \( 1 + (-0.0801 + 0.356i)T + (-64.1 - 30.4i)T^{2} \) |
| 73 | \( 1 + (-3.64 + 5.28i)T + (-25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (-8.35 + 3.16i)T + (59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (6.38 + 12.1i)T + (-47.1 + 68.3i)T^{2} \) |
| 89 | \( 1 + (-1.09 - 0.293i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-14.7 + 4.92i)T + (77.5 - 58.2i)T^{2} \) |
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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
−9.906150296863167964680660652267, −9.501390414044782060771768180244, −8.738371650550756327557928287012, −8.164077610862770041921641533634, −7.69973302415578388457220503767, −6.16501748799965351213941852995, −5.38422297667135808912021073369, −4.49788121027974501129292794071, −3.39803389916223757881196158956, −1.81094567541662160179107737888,
0.960349546674912580088330080690, 2.24893032975519566601356182109, 2.68252776827949337721347527974, 3.57659817321304466896412134758, 5.34980462006207341880970740143, 6.75299469019979830820952594013, 7.55658957402794978030219513366, 8.117340620759471222300383140277, 9.115163621652285290483646274101, 9.904405961378470749812982835601
