L(s) = 1 | + (−1.22 + 1.22i)2-s + (−1.22 + 1.22i)3-s + 1.00i·4-s − 5i·5-s − 2.99i·6-s − 9.79i·7-s + (−6.12 − 6.12i)8-s − 2.99i·9-s + (6.12 + 6.12i)10-s + (7 + 7i)11-s + (−1.22 − 1.22i)12-s − 9.79·13-s + (11.9 + 11.9i)14-s + (6.12 + 6.12i)15-s + 10.9·16-s + (−14.6 + 14.6i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.612i)2-s + (−0.408 + 0.408i)3-s + 0.250i·4-s − i·5-s − 0.499i·6-s − 1.39i·7-s + (−0.765 − 0.765i)8-s − 0.333i·9-s + (0.612 + 0.612i)10-s + (0.636 + 0.636i)11-s + (−0.102 − 0.102i)12-s − 0.753·13-s + (0.857 + 0.857i)14-s + (0.408 + 0.408i)15-s + 0.687·16-s + (−0.864 + 0.864i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 - 0.560i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.159309 + 0.519466i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.159309 + 0.519466i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 + 5iT \) |
| 29 | \( 1 - 29iT \) |
good | 2 | \( 1 + (1.22 - 1.22i)T - 4iT^{2} \) |
| 7 | \( 1 + 9.79iT - 49T^{2} \) |
| 11 | \( 1 + (-7 - 7i)T + 121iT^{2} \) |
| 13 | \( 1 + 9.79T + 169T^{2} \) |
| 17 | \( 1 + (14.6 - 14.6i)T - 289iT^{2} \) |
| 19 | \( 1 + (-11 - 11i)T + 361iT^{2} \) |
| 23 | \( 1 - 39.1iT - 529T^{2} \) |
| 31 | \( 1 + (11 + 11i)T + 961iT^{2} \) |
| 37 | \( 1 + (-34.2 - 34.2i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (-5 + 5i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (-34.2 + 34.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (53.8 + 53.8i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 - 88.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 24T + 3.48e3T^{2} \) |
| 61 | \( 1 + (-49 - 49i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + 19.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 130iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-4.89 - 4.89i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + (-11 - 11i)T + 6.24e3iT^{2} \) |
| 83 | \( 1 - 117. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (77 + 77i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (14.6 + 14.6i)T + 9.40e3iT^{2} \) |
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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
−11.27463329582049737584478218306, −10.00029031385712749056741976565, −9.538654524798715565806794804287, −8.564726784224374771675953366957, −7.52745813602569857297404416036, −6.96142406469954243033108593438, −5.67935697449589705013065381310, −4.39334813567143341052258339934, −3.73666687255354073321542677441, −1.24929214608718925687120346939,
0.32211349937440583287848206763, 2.22020773349486801716539991456, 2.81353355050407168123183194091, 4.89819487392292032318352565035, 6.04256466901313139372924069663, 6.60146291618557841726668268475, 7.946662784305389285529229873746, 9.074131121200664241857319271765, 9.596595386837690868001305695584, 10.76421258719239764738967890641