L(s) = 1 | + (−1.08 − 0.909i)2-s + (0.345 + 1.96i)4-s + (−0.780 + 2.09i)5-s + (−2.10 − 2.10i)7-s + (1.41 − 2.44i)8-s + (2.75 − 1.55i)10-s + 3.11·11-s + (−2.19 + 2.19i)13-s + (0.366 + 4.20i)14-s + (−3.76 + 1.36i)16-s + (−5.48 + 5.48i)17-s + 3.91i·19-s + (−4.39 − 0.813i)20-s + (−3.37 − 2.83i)22-s + (−2.42 + 2.42i)23-s + ⋯ |
L(s) = 1 | + (−0.765 − 0.643i)2-s + (0.172 + 0.984i)4-s + (−0.349 + 0.937i)5-s + (−0.796 − 0.796i)7-s + (0.500 − 0.865i)8-s + (0.869 − 0.493i)10-s + 0.940·11-s + (−0.608 + 0.608i)13-s + (0.0978 + 1.12i)14-s + (−0.940 + 0.340i)16-s + (−1.33 + 1.33i)17-s + 0.899i·19-s + (−0.983 − 0.181i)20-s + (−0.720 − 0.604i)22-s + (−0.505 + 0.505i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.296842 + 0.340203i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.296842 + 0.340203i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.08 + 0.909i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.780 - 2.09i)T \) |
good | 7 | \( 1 + (2.10 + 2.10i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.11T + 11T^{2} \) |
| 13 | \( 1 + (2.19 - 2.19i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.48 - 5.48i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.91iT - 19T^{2} \) |
| 23 | \( 1 + (2.42 - 2.42i)T - 23iT^{2} \) |
| 29 | \( 1 + 2.23T + 29T^{2} \) |
| 31 | \( 1 - 7.17iT - 31T^{2} \) |
| 37 | \( 1 + (-1.23 - 1.23i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.40T + 41T^{2} \) |
| 43 | \( 1 + (1.50 + 1.50i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.00 - 2.00i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.56 + 5.56i)T - 53iT^{2} \) |
| 59 | \( 1 + 5.44iT - 59T^{2} \) |
| 61 | \( 1 - 7.46iT - 61T^{2} \) |
| 67 | \( 1 + (-6.40 + 6.40i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.00iT - 71T^{2} \) |
| 73 | \( 1 + (1.49 + 1.49i)T + 73iT^{2} \) |
| 79 | \( 1 - 1.91T + 79T^{2} \) |
| 83 | \( 1 + (-6.67 - 6.67i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.0iT - 89T^{2} \) |
| 97 | \( 1 + (10.0 - 10.0i)T - 97iT^{2} \) |
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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
−11.54376028266928904522869166287, −10.60965961135222999203158295795, −10.05715474102669231480987901670, −9.100049073106131558043633179254, −8.019802998924720848628738055157, −6.91441875357068584637621892460, −6.49419425928900522976346106161, −4.08849152237613092542789348326, −3.51321186942508232967767308597, −1.91913906810696340643963994584,
0.37965394699664438392654909508, 2.43029026753941095511293994149, 4.42857044187458427487236606607, 5.46295731524143212612514906181, 6.51311061630151311899109823453, 7.43766956491780047619414149670, 8.628409761373963520053093882259, 9.206979054250768515948526484022, 9.793938990298187340434411271622, 11.25206692105713354847646602249