L(s) = 1 | + 3i·3-s + (−2 − i)5-s − i·7-s − 6·9-s − 3·11-s − i·13-s + (3 − 6i)15-s − 5i·17-s − 8·19-s + 3·21-s + 2i·23-s + (3 + 4i)25-s − 9i·27-s + 29-s + 2·31-s + ⋯ |
L(s) = 1 | + 1.73i·3-s + (−0.894 − 0.447i)5-s − 0.377i·7-s − 2·9-s − 0.904·11-s − 0.277i·13-s + (0.774 − 1.54i)15-s − 1.21i·17-s − 1.83·19-s + 0.654·21-s + 0.417i·23-s + (0.600 + 0.800i)25-s − 1.73i·27-s + 0.185·29-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - 3iT - 3T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + 5iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 11iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 7T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 3iT - 97T^{2} \) |
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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
−10.39122221674609264460421941462, −9.813340107754439583591978097041, −8.673555457906657883500618249836, −8.199690345447704683467608904797, −6.90625967382598233767942439212, −5.36047721508849218345547444582, −4.72136855038995652583907868073, −3.91813131858336118609679447058, −2.89313021270953584683833638981, 0,
1.90029477991754274331086169687, 2.90182251378309740686117137120, 4.40118322555487914118174305000, 6.00644602808738444314514610434, 6.54283079933294675299992408879, 7.58129235973462725917552446622, 8.129436152953682363906960379319, 8.835539964443643956614724393747, 10.58804003912029222761714093689