L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + 0.999i·6-s + (−2.37 − 4.12i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s − 5.36·11-s + (0.499 − 0.866i)12-s + (−1.67 + 0.968i)13-s + 4.75i·14-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s + (3.25 + 1.87i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + 0.408i·6-s + (−0.899 − 1.55i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + 0.316·10-s − 1.61·11-s + (0.144 − 0.249i)12-s + (−0.465 + 0.268i)13-s + 1.27i·14-s + (0.223 + 0.129i)15-s + (−0.125 + 0.216i)16-s + (0.789 + 0.455i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3071558488\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3071558488\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 37 | \( 1 + (-2.60 + 5.49i)T \) |
good | 7 | \( 1 + (2.37 + 4.12i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 5.36T + 11T^{2} \) |
| 13 | \( 1 + (1.67 - 0.968i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.25 - 1.87i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.63 + 0.945i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 0.204iT - 23T^{2} \) |
| 29 | \( 1 - 2.07iT - 29T^{2} \) |
| 31 | \( 1 - 1.95iT - 31T^{2} \) |
| 41 | \( 1 + (-3.63 - 6.29i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 1.74iT - 43T^{2} \) |
| 47 | \( 1 - 8.35T + 47T^{2} \) |
| 53 | \( 1 + (6.35 - 10.9i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (8.23 + 4.75i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.60 - 2.07i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.26 + 3.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.89 - 8.47i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + (10.1 - 5.88i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.10 - 3.64i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.32 - 3.07i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2.30iT - 97T^{2} \) |
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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
−10.14448663368817007001663158162, −9.331177789487980024429068482409, −7.972562259833921603034842483641, −7.55618361408044278284836860974, −6.95971300876286078041460766302, −5.91326745457609025289615361275, −4.64295055116221871851464865799, −3.51362307114932215959367520624, −2.62313325565697679670792683724, −0.961064926750539752848348171426,
0.21349065529766710323582832363, 2.44205441632028104908802624983, 3.23665260337974752985247418877, 4.91135253100472608198785927103, 5.52124490521754299047031018879, 6.19021983968369704985879771970, 7.47630180897819332556131729340, 8.104977337888073380152857350805, 9.017030848971301357470038611396, 9.698045715729860655188915451566