L(s) = 1 | + (−1 − 2i)5-s − 2i·7-s + 2i·11-s + 2·13-s + 4i·17-s + 2i·19-s + 6i·23-s + (−3 + 4i)25-s − 4i·29-s + 8·31-s + (−4 + 2i)35-s + 10·37-s + 6·41-s + 4·43-s + 6i·47-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.894i)5-s − 0.755i·7-s + 0.603i·11-s + 0.554·13-s + 0.970i·17-s + 0.458i·19-s + 1.25i·23-s + (−0.600 + 0.800i)25-s − 0.742i·29-s + 1.43·31-s + (−0.676 + 0.338i)35-s + 1.64·37-s + 0.937·41-s + 0.609·43-s + 0.875i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.684269208\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.684269208\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−8.642625844249209508787148403382, −7.82463194663249580365793420036, −7.58155860841357234699511739885, −6.33234118097234750810990682118, −5.75346998660951441759653829124, −4.51964840539373667004252929861, −4.22179773891706083744489012477, −3.24544002672237736098148261049, −1.77787049368792626157617354394, −0.858207523796355079723462412511,
0.77875567907447025365915348453, 2.55061697328119929627044565661, 2.88949821771589606266114847200, 4.04225869926239842134364107290, 4.88028910859973911356463579475, 5.98514380004466445770661890363, 6.42462164491386394739469861875, 7.31654788048108036590523117773, 8.067914924638430663093665063319, 8.790429662327969257560757173359