L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−1.44 − 1.44i)7-s + (−0.707 − 0.707i)8-s − 1.09i·11-s + (−4.22 + 4.22i)13-s − 2.04·14-s − 1.00·16-s + (−4.17 + 4.17i)17-s − 4.44i·19-s + (−0.775 − 0.775i)22-s + (−4.48 − 4.48i)23-s + 5.97i·26-s + (−1.44 + 1.44i)28-s − 3.14·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.547 − 0.547i)7-s + (−0.250 − 0.250i)8-s − 0.330i·11-s + (−1.17 + 1.17i)13-s − 0.547·14-s − 0.250·16-s + (−1.01 + 1.01i)17-s − 1.02i·19-s + (−0.165 − 0.165i)22-s + (−0.936 − 0.936i)23-s + 1.17i·26-s + (−0.273 + 0.273i)28-s − 0.584·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3326790155\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3326790155\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.44 + 1.44i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.09iT - 11T^{2} \) |
| 13 | \( 1 + (4.22 - 4.22i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.17 - 4.17i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.44iT - 19T^{2} \) |
| 23 | \( 1 + (4.48 + 4.48i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.14T + 29T^{2} \) |
| 31 | \( 1 + 1.44T + 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 4.87iT - 41T^{2} \) |
| 43 | \( 1 + (-7.22 + 7.22i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.31 - 7.31i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.65 - 5.65i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 8.44T + 61T^{2} \) |
| 67 | \( 1 + (2 + 2i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.9iT - 71T^{2} \) |
| 73 | \( 1 + (-4.44 + 4.44i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.44iT - 79T^{2} \) |
| 83 | \( 1 + (10.2 + 10.2i)T + 83iT^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 + (8.44 + 8.44i)T + 97iT^{2} \) |
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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
−9.297367875944315864796157576822, −8.502786158511273700420321370071, −7.24262800342242130744667719381, −6.64865171109052875565995419753, −5.79481037139119385957751478872, −4.50117465147004223127474515208, −4.13371799627801990896111479190, −2.83293184486740103251920899154, −1.88880807803154799783649915003, −0.10391638168134712855046361087,
2.19567197903101455895172194495, 3.12340677172722135203462579153, 4.19010504098864344940238571729, 5.26730328501256514795857130585, 5.77594108562396818945248199025, 6.82951097662907495931920402533, 7.55548587274738456502004049342, 8.261030660053741988513316894528, 9.414313615680439383326790817887, 9.804157097672539544078856161106