L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.133 − 2.23i)5-s + (−1.73 + 2i)7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (1.23 − 1.86i)10-s + (−1.5 − 2.59i)11-s − 5i·13-s + (−2.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (1.73 − i)17-s + (−2.59 + 1.5i)18-s + (−2.5 + 4.33i)19-s + (1.99 − 0.999i)20-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.0599 − 0.998i)5-s + (−0.654 + 0.755i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (0.389 − 0.590i)10-s + (−0.452 − 0.783i)11-s − 1.38i·13-s + (−0.668 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.420 − 0.242i)17-s + (−0.612 + 0.353i)18-s + (−0.573 + 0.993i)19-s + (0.447 − 0.223i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13429 + 0.192898i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13429 + 0.192898i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-0.133 + 2.23i)T \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
good | 3 | \( 1 + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.06 - 3.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + (6.06 + 3.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.79 + 4.5i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (13.8 - 8i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−14.82997562510298714300094279101, −13.41854181397001313406656724067, −12.88921656304410320616470455147, −11.78579503370437776568950278005, −10.33937039601442105215947901050, −8.732000280773706090483137127007, −7.897795448188747920263290077983, −5.86277139977059120087424334488, −5.17048229971270943599787253056, −3.04827988601823267908083379659,
2.84781537432164549982239978598, 4.32092512076614076650127781573, 6.37698392419049473313456733974, 7.08691500602834560351759685181, 9.257965064754940141499293867865, 10.36885151170820898870890211826, 11.34454649650192893702382841885, 12.51376304695775317352337517652, 13.63848478873462574957290436461, 14.57856116501803666049310419758