L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−1.56 − 1.59i)5-s + 0.999i·6-s + (−0.998 + 0.576i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−2.16 + 0.553i)10-s + 1.60·11-s + (0.866 + 0.499i)12-s + (−1.89 − 3.28i)13-s + 1.15i·14-s + (2.15 + 0.603i)15-s + (−0.5 + 0.866i)16-s + (−3.28 + 5.69i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.698 − 0.715i)5-s + 0.408i·6-s + (−0.377 + 0.217i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.685 + 0.175i)10-s + 0.482·11-s + (0.249 + 0.144i)12-s + (−0.526 − 0.911i)13-s + 0.308i·14-s + (0.555 + 0.155i)15-s + (−0.125 + 0.216i)16-s + (−0.797 + 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0878 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0878 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3751662796\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3751662796\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (1.56 + 1.59i)T \) |
| 37 | \( 1 + (6.07 + 0.267i)T \) |
good | 7 | \( 1 + (0.998 - 0.576i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 1.60T + 11T^{2} \) |
| 13 | \( 1 + (1.89 + 3.28i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.28 - 5.69i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.174 - 0.100i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 0.633T + 23T^{2} \) |
| 29 | \( 1 + 5.73iT - 29T^{2} \) |
| 31 | \( 1 - 8.34iT - 31T^{2} \) |
| 41 | \( 1 + (-4.00 - 6.93i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 1.95T + 43T^{2} \) |
| 47 | \( 1 - 4.65iT - 47T^{2} \) |
| 53 | \( 1 + (-7.47 - 4.31i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.3 - 5.99i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (11.3 - 6.54i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.33 - 5.38i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.52 + 9.56i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 4.32iT - 73T^{2} \) |
| 79 | \( 1 + (-6.91 + 3.99i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (12.4 + 7.21i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-15.6 - 9.02i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2.93T + 97T^{2} \) |
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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
−10.29562305606379227763842872228, −9.241076349258914141656733626287, −8.626807567993831346610480832836, −7.62634592472881818419707540179, −6.43460231521877516713982666172, −5.63488704290274233721793964851, −4.66768875546494421465946957290, −4.00185579553885432330131946281, −2.95822623308380052902020016806, −1.35010183261987820898663277644,
0.16376100877268086375514783220, 2.34723461027130848958449279108, 3.61005752239806984410130333558, 4.45876349396332873652486239085, 5.40283034269369920910524205118, 6.73068872621993696906494077171, 6.83500083057529703187410123424, 7.61495995468182641671797185774, 8.760139719620071694286598675363, 9.556061502641009567573550815678