L(s) = 1 | + (0.750 − 1.19i)2-s + (−0.874 − 1.79i)4-s + (−1.07 − 1.95i)5-s − 1.22·7-s + (−2.81 − 0.302i)8-s + (−3.15 − 0.178i)10-s + (−1.38 − 1.38i)11-s + (2.12 + 2.12i)13-s + (−0.916 + 1.46i)14-s + (−2.47 + 3.14i)16-s − 6.00i·17-s + (−3.06 + 3.06i)19-s + (−2.58 + 3.65i)20-s + (−2.69 + 0.619i)22-s − 2.90·23-s + ⋯ |
L(s) = 1 | + (0.530 − 0.847i)2-s + (−0.437 − 0.899i)4-s + (−0.481 − 0.876i)5-s − 0.461·7-s + (−0.994 − 0.106i)8-s + (−0.998 − 0.0565i)10-s + (−0.416 − 0.416i)11-s + (0.588 + 0.588i)13-s + (−0.244 + 0.391i)14-s + (−0.618 + 0.786i)16-s − 1.45i·17-s + (−0.702 + 0.702i)19-s + (−0.577 + 0.816i)20-s + (−0.574 + 0.132i)22-s − 0.606·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.244861 + 0.857573i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.244861 + 0.857573i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.750 + 1.19i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.07 + 1.95i)T \) |
good | 7 | \( 1 + 1.22T + 7T^{2} \) |
| 11 | \( 1 + (1.38 + 1.38i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.12 - 2.12i)T + 13iT^{2} \) |
| 17 | \( 1 + 6.00iT - 17T^{2} \) |
| 19 | \( 1 + (3.06 - 3.06i)T - 19iT^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 + (3.18 - 3.18i)T - 29iT^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 + (-2.44 + 2.44i)T - 37iT^{2} \) |
| 41 | \( 1 + 2.38iT - 41T^{2} \) |
| 43 | \( 1 + (-9.00 + 9.00i)T - 43iT^{2} \) |
| 47 | \( 1 + 0.586iT - 47T^{2} \) |
| 53 | \( 1 + (-2.36 + 2.36i)T - 53iT^{2} \) |
| 59 | \( 1 + (8.43 + 8.43i)T + 59iT^{2} \) |
| 61 | \( 1 + (-9.98 + 9.98i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.82 + 3.82i)T + 67iT^{2} \) |
| 71 | \( 1 - 11.5iT - 71T^{2} \) |
| 73 | \( 1 - 1.31T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + (-2.91 - 2.91i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.58iT - 89T^{2} \) |
| 97 | \( 1 + 9.45iT - 97T^{2} \) |
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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
−9.928701970552063464421815639732, −9.163775585657611508668770279879, −8.490514974716141921368748112107, −7.26309108374772066041400336360, −6.01143360115785250721187011261, −5.21474164390484698879519889144, −4.18193347259660926620894614043, −3.38609937358278156314724185140, −1.95024857188398519250226942901, −0.37383087944712621353486974065,
2.59184618112804488074151507408, 3.65778967417546125712416629399, 4.44103313761794952034641497520, 5.92206663922050400095771278748, 6.33689451950451224071250390338, 7.45336967704280854315975422277, 8.006239830482199346464102130262, 8.970589245580943821160764024413, 10.12888966917850665021196434152, 10.90345885846796250703668654997