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 + (2.23 + 3.86i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s + 2·11-s + (−0.499 + 0.866i)12-s + (−0.866 + 0.5i)13-s + 4.46i·14-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s + (−1.26 − 0.732i)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 + (0.843 + 1.46i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + 0.316·10-s + 0.603·11-s + (−0.144 + 0.249i)12-s + (−0.240 + 0.138i)13-s + 1.19i·14-s + (0.223 + 0.129i)15-s + (−0.125 + 0.216i)16-s + (−0.307 − 0.177i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.923000336\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.923000336\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.59 - 5.5i)T \) |
good | 7 | \( 1 + (-2.23 - 3.86i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.26 + 0.732i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.59 - 1.5i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4.73iT - 23T^{2} \) |
| 29 | \( 1 + 3.73iT - 29T^{2} \) |
| 31 | \( 1 + 5.46iT - 31T^{2} \) |
| 41 | \( 1 + (-2.09 - 3.63i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 2.19iT - 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 + (0.366 - 0.633i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.63 + 2.09i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.02 - 4.63i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.73 - 3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.401 - 0.696i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 5.80T + 73T^{2} \) |
| 79 | \( 1 + (-0.169 + 0.0980i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.96 + 8.59i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.19 + 1.26i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 15.1iT - 97T^{2} \) |
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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
−9.916910322655782158353552091135, −9.050004117741158666827168718926, −8.523817565852962375295648902382, −7.72702665898368536917818090295, −6.37613647204296684491145733113, −5.80482597388852968616672723722, −4.81981029267178960552202250848, −4.23390965597585915922729339663, −2.76449855482346181568430756024, −1.97599574757692054837633050136,
1.10702026584507722750464599558, 2.06865745465827218817956831358, 3.41589326525930644051298024250, 4.25043746480939575025856416942, 5.18074505051056525125209611008, 6.33766179214080872604898434387, 7.10291904435687539948480390299, 7.72091566870909207429672893814, 8.844157754371529533515450826089, 9.724667354900275850984228861083