L(s) = 1 | + (0.366 + 1.36i)2-s + (−1.73 + i)4-s + (0.133 + 2.23i)5-s + (−2 − 1.99i)8-s + (−2.59 + 1.5i)9-s + (−2.99 + i)10-s + (1.59 + 3.23i)13-s + (1.99 − 3.46i)16-s + (0.0358 − 0.133i)17-s + (−3 − 3i)18-s + (−2.46 − 3.73i)20-s + (−4.96 + 0.598i)25-s + (−3.83 + 3.36i)26-s + (9.23 + 5.33i)29-s + (5.46 + 1.46i)32-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.5i)4-s + (0.0599 + 0.998i)5-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (−0.948 + 0.316i)10-s + (0.443 + 0.896i)13-s + (0.499 − 0.866i)16-s + (0.00870 − 0.0324i)17-s + (−0.707 − 0.707i)18-s + (−0.550 − 0.834i)20-s + (−0.992 + 0.119i)25-s + (−0.751 + 0.660i)26-s + (1.71 + 0.989i)29-s + (0.965 + 0.258i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.273802 + 1.10626i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.273802 + 1.10626i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 - 1.36i)T \) |
| 5 | \( 1 + (-0.133 - 2.23i)T \) |
| 13 | \( 1 + (-1.59 - 3.23i)T \) |
good | 3 | \( 1 + (2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.0358 + 0.133i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-9.23 - 5.33i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (2.86 + 10.6i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-10.3 - 5.96i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (-5.29 + 5.29i)T - 53iT^{2} \) |
| 59 | \( 1 + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.33 - 2.30i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.16 + 1.16i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (17.7 + 4.75i)T + (84.0 + 48.5i)T^{2} \) |
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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
−12.50085451392675585442148941756, −11.43204248981908133472460572684, −10.53145971876565878089942232029, −9.272910550970650109539882760694, −8.371000029517526354214094418073, −7.30584786406429692685882214829, −6.44201946370872960518714546572, −5.50659100733429146511757047446, −4.12372830199504256312742633006, −2.78129623757159189916483365992,
0.866992351343229496059516783790, 2.74673051334977702377817061566, 4.07024148128909104046173651961, 5.25933939926732076545641023941, 6.12740126535897525430798517549, 8.166095409120280519768180337559, 8.772771927299445725384838735213, 9.772843918760856572033721003635, 10.71031258275367892220772446309, 11.84033389376177807957909186058