L(s) = 1 | + i·3-s + i·7-s − 9-s − i·13-s − 3·19-s − 21-s − 4i·23-s − i·27-s + 4·29-s + 7·31-s − 6i·37-s + 39-s + 6·41-s + 9i·43-s + 6i·47-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 0.277i·13-s − 0.688·19-s − 0.218·21-s − 0.834i·23-s − 0.192i·27-s + 0.742·29-s + 1.25·31-s − 0.986i·37-s + 0.160·39-s + 0.937·41-s + 1.37i·43-s + 0.875i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.755419582\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.755419582\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 18iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 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
−8.316135278026738864693456958040, −7.970949020447044529839592132840, −6.81714645204455009198314030039, −6.20814052585699346746704718872, −5.47128315572202408837661689058, −4.59940614615979281286986474824, −4.07195411272356769426395045363, −2.93993092537491935157029145286, −2.34094080351925932207094068322, −0.872304012292497807276476779638,
0.62336955883228869679325202408, 1.70142717203820727821630993050, 2.62487427599924820014779532100, 3.57878769141661893507591496728, 4.44048838516414432484857698573, 5.22480626763150479651552833548, 6.19636988814089754960505348976, 6.65672175128741560907409862380, 7.47092413132245305856124883170, 8.051961879923222673077970711170