L(s) = 1 | + (−1.16 + 0.312i)3-s + (−0.121 − 2.23i)5-s + (−2.59 + 0.491i)7-s + (−1.33 + 0.770i)9-s + (−1.67 + 2.90i)11-s + (−2.92 − 2.92i)13-s + (0.840 + 2.56i)15-s + (0.0694 + 0.259i)17-s + (−0.458 − 0.793i)19-s + (2.88 − 1.38i)21-s + (−7.58 − 2.03i)23-s + (−4.97 + 0.544i)25-s + (3.87 − 3.87i)27-s − 1.31i·29-s + (3.25 + 1.88i)31-s + ⋯ |
L(s) = 1 | + (−0.673 + 0.180i)3-s + (−0.0544 − 0.998i)5-s + (−0.982 + 0.185i)7-s + (−0.444 + 0.256i)9-s + (−0.506 + 0.877i)11-s + (−0.810 − 0.810i)13-s + (0.216 + 0.662i)15-s + (0.0168 + 0.0628i)17-s + (−0.105 − 0.182i)19-s + (0.628 − 0.302i)21-s + (−1.58 − 0.423i)23-s + (−0.994 + 0.108i)25-s + (0.746 − 0.746i)27-s − 0.244i·29-s + (0.584 + 0.337i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00800376 - 0.117399i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00800376 - 0.117399i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.121 + 2.23i)T \) |
| 7 | \( 1 + (2.59 - 0.491i)T \) |
good | 3 | \( 1 + (1.16 - 0.312i)T + (2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (1.67 - 2.90i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.92 + 2.92i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.0694 - 0.259i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.458 + 0.793i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.58 + 2.03i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 1.31iT - 29T^{2} \) |
| 31 | \( 1 + (-3.25 - 1.88i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.11 + 7.90i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (2.61 - 2.61i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.911 + 0.244i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.49 - 13.0i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.91 + 6.78i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.70 + 5.60i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.37 + 1.44i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + (7.33 - 1.96i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (1.87 - 1.08i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.49 + 5.49i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.34 + 2.33i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.1 - 10.1i)T - 97iT^{2} \) |
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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
−11.58590384254675212806379538573, −10.24282388807917355312740812002, −9.774341513755168758214986679861, −8.527000801951196131451000079722, −7.57240351090141854363694996280, −6.14734612671133020999772680629, −5.32283508686170750002523775752, −4.33361059966823359466793177094, −2.54019345114765807860554968518, −0.088655455158749105155736033102,
2.65354313945599222190451924259, 3.81614485961214514550032329993, 5.58533548503462767514115918440, 6.38542809131578359171148196613, 7.14916940097486162230107952942, 8.427181783041623248803428528912, 9.786557514963137011497931115416, 10.39516690350320179099335498478, 11.56739835138239315228329870403, 11.95397900606036234983498916198