L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s − i·5-s + (0.866 − 0.499i)6-s + (1.73 − i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)10-s + (1.73 + i)11-s + 0.999·12-s + 1.99·14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (2.5 + 4.33i)17-s − 0.999i·18-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s − 0.447i·5-s + (0.353 − 0.204i)6-s + (0.654 − 0.377i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (0.522 + 0.301i)11-s + 0.288·12-s + 0.534·14-s + (−0.223 − 0.129i)15-s + (−0.125 + 0.216i)16-s + (0.606 + 1.05i)17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.786634438\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.786634438\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + iT - 5T^{2} \) |
| 7 | \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.73 - i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.73 - i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + (9.52 + 5.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.33 + 2.5i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + T + 53T^{2} \) |
| 59 | \( 1 + (-6.92 + 4i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.73 + i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (12.1 - 7i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 13iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + (-1.73 - i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.73 - i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.944495794220876967538892601410, −8.677964685794439686867458266901, −8.283966009328387614924058762009, −7.33909063883004444667993586738, −6.55120266792438192233634043704, −5.66003858693035443548859144301, −4.56605556387738202044227674248, −3.88870813272920274318934168802, −2.48794050047420119202186078314, −1.23580470480409717011374851339,
1.50465139085063297129044890445, 2.91516793863115381196224805964, 3.51264861345915470364509317338, 4.90091955475211292105965876150, 5.25916793241391636916524349854, 6.60001802584233314163483371562, 7.32299905648529210536018500907, 8.597675431340149996085705777482, 9.118034131451334464392980819770, 10.23453457476844323243449701342