| L(s) = 1 | + (−1.37 + 0.328i)2-s + (−1.63 − 0.324i)3-s + (1.78 − 0.902i)4-s + (0.608 − 2.15i)5-s + (2.35 − 0.0889i)6-s + (−0.693 − 0.463i)7-s + (−2.15 + 1.82i)8-s + (−0.211 − 0.0876i)9-s + (−0.131 + 3.15i)10-s + (5.70 − 1.13i)11-s + (−3.20 + 0.894i)12-s + (−0.791 − 0.791i)13-s + (1.10 + 0.409i)14-s + (−1.69 + 3.31i)15-s + (2.37 − 3.22i)16-s + (1.86 − 3.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.972 + 0.232i)2-s + (−0.942 − 0.187i)3-s + (0.892 − 0.451i)4-s + (0.272 − 0.962i)5-s + (0.960 − 0.0363i)6-s + (−0.261 − 0.175i)7-s + (−0.763 + 0.646i)8-s + (−0.0705 − 0.0292i)9-s + (−0.0415 + 0.999i)10-s + (1.71 − 0.341i)11-s + (−0.925 + 0.258i)12-s + (−0.219 − 0.219i)13-s + (0.295 + 0.109i)14-s + (−0.437 + 0.855i)15-s + (0.592 − 0.805i)16-s + (0.451 − 0.892i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.271651 - 0.499363i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.271651 - 0.499363i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 0.328i)T \) |
| 5 | \( 1 + (-0.608 + 2.15i)T \) |
| 17 | \( 1 + (-1.86 + 3.67i)T \) |
| good | 3 | \( 1 + (1.63 + 0.324i)T + (2.77 + 1.14i)T^{2} \) |
| 7 | \( 1 + (0.693 + 0.463i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-5.70 + 1.13i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (0.791 + 0.791i)T + 13iT^{2} \) |
| 19 | \( 1 + (-1.74 + 0.720i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.716 - 3.59i)T + (-21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (7.13 - 4.76i)T + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + (-0.420 + 2.11i)T + (-28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-0.424 + 2.13i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (3.08 + 2.06i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-1.72 + 4.16i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (7.04 + 7.04i)T + 47iT^{2} \) |
| 53 | \( 1 + (-8.10 + 3.35i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (4.00 - 9.67i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (8.90 + 5.94i)T + (23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 - 9.22T + 67T^{2} \) |
| 71 | \( 1 + (2.73 + 0.544i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (9.15 + 13.7i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (0.721 + 3.62i)T + (-72.9 + 30.2i)T^{2} \) |
| 83 | \( 1 + (1.61 - 0.670i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (8.32 - 8.32i)T - 89iT^{2} \) |
| 97 | \( 1 + (-1.19 + 0.801i)T + (37.1 - 89.6i)T^{2} \) |
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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
−9.991657943088671699752999726905, −9.230408541271241961532291905431, −8.767055634809185757003758503481, −7.46623773483797524106868136243, −6.69598934251282837318031765042, −5.77800414632750702739823066287, −5.15748973273100786758402772222, −3.48645105734050435514248398458, −1.56089193343719328748616544013, −0.49895783477172579516181650727,
1.54837931626423662712020217991, 2.96256528270353586529578419997, 4.14475833436995364661919495433, 5.88126047123571665300265916235, 6.36867483219786538150933329343, 7.14691479015162166521000741700, 8.259281246263905613681930190936, 9.396028818986008264828839527201, 9.901901490274098767703420371775, 10.79634675382149522811532425132
