L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.75 + 3.03i)5-s + (1.12 − 2.39i)7-s − 0.999·8-s − 3.50·10-s + 6.40·11-s + (−0.213 − 3.59i)13-s + (2.63 − 0.222i)14-s + (−0.5 − 0.866i)16-s + (2.33 − 4.05i)17-s + 5.23·19-s + (−1.75 − 3.03i)20-s + (3.20 + 5.54i)22-s + (−1.08 − 1.87i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.784 + 1.35i)5-s + (0.425 − 0.904i)7-s − 0.353·8-s − 1.10·10-s + 1.93·11-s + (−0.0590 − 0.998i)13-s + (0.704 − 0.0593i)14-s + (−0.125 − 0.216i)16-s + (0.567 − 0.982i)17-s + 1.20·19-s + (−0.392 − 0.679i)20-s + (0.683 + 1.18i)22-s + (−0.226 − 0.391i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.096871958\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.096871958\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.12 + 2.39i)T \) |
| 13 | \( 1 + (0.213 + 3.59i)T \) |
good | 5 | \( 1 + (1.75 - 3.03i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 - 6.40T + 11T^{2} \) |
| 17 | \( 1 + (-2.33 + 4.05i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 5.23T + 19T^{2} \) |
| 23 | \( 1 + (1.08 + 1.87i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.23 + 2.13i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.46 - 7.72i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.94 + 3.37i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.09 - 8.82i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.19 + 2.06i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.44 + 4.22i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.05 - 1.82i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.89 - 10.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 9.35T + 61T^{2} \) |
| 67 | \( 1 - 7.28T + 67T^{2} \) |
| 71 | \( 1 + (2.79 + 4.83i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.23 - 7.32i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.893 + 1.54i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.59T + 83T^{2} \) |
| 89 | \( 1 + (-3.50 - 6.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.92 + 8.53i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.572083481487341655603970315088, −8.391024778641377093941021865130, −7.70167988227920934393143321912, −6.92712498287281473506894057477, −6.70249915464998013070603729918, −5.45542931456740730405828974482, −4.40070715639992985467743357978, −3.57645286446673687982105978645, −3.00271036692506920273408367098, −0.988405058395318234240666943318,
1.10747891749745251526495632088, 1.84803938097210401635074452884, 3.53825876026106760450690451347, 4.14636289668335164700339177837, 4.94027807874224611793182475672, 5.77577856872778814805783123369, 6.71814884988443759271261557665, 7.973796523542100442215484812596, 8.615479803277991626045068206062, 9.288617664672834866012611740248