L(s) = 1 | − 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s − 8-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (−1 − 1.73i)13-s + (0.5 − 0.866i)14-s + 0.999·15-s + 16-s + (−1.64 + 2.85i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.288 + 0.499i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (0.204 − 0.353i)6-s + (−0.188 + 0.327i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 + 0.273i)10-s + (0.150 + 0.261i)11-s + (−0.144 + 0.249i)12-s + (−0.277 − 0.480i)13-s + (0.133 − 0.231i)14-s + 0.258·15-s + 0.250·16-s + (−0.399 + 0.691i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.126084 - 0.241474i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.126084 - 0.241474i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-1.14 + 5.44i)T \) |
good | 7 | \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.64 - 2.85i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.64 + 2.85i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 3.29T + 23T^{2} \) |
| 29 | \( 1 + 6.29T + 29T^{2} \) |
| 37 | \( 1 + (0.645 - 1.11i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.64 + 4.58i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 3.29T + 61T^{2} \) |
| 67 | \( 1 + (0.645 + 1.11i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.29 + 12.6i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.29 + 3.96i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.14 + 7.18i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−9.671664425722558721652465550562, −9.099826071169551749483190121421, −8.217761540986552364971056994411, −7.40310846588771260934708763319, −6.32620493763640991813724636748, −5.49137040459936073345599428862, −4.43960625838953508121481906515, −3.32610337519141172966699550193, −1.94110801735605350955286700965, −0.16675286034182692367101090418,
1.46868765705065528659553402138, 2.76887853579925944867846487861, 3.96038026120998768426769521792, 5.32211297424046772191797192797, 6.38353107943677925950269163073, 7.04358485733164412061460734172, 7.75687410178043076140792760574, 8.626454953602248949349512372140, 9.608077082042109913936668968972, 10.26662358131819448033600623916