L(s) = 1 | − i·7-s − 3·11-s − 4i·13-s − 2·19-s + 3i·23-s + 9·29-s + 8·31-s − 5i·37-s + 6·41-s + 11i·43-s + 6i·47-s − 49-s − 6i·53-s − 10·61-s − 5i·67-s + ⋯ |
L(s) = 1 | − 0.377i·7-s − 0.904·11-s − 1.10i·13-s − 0.458·19-s + 0.625i·23-s + 1.67·29-s + 1.43·31-s − 0.821i·37-s + 0.937·41-s + 1.67i·43-s + 0.875i·47-s − 0.142·49-s − 0.824i·53-s − 1.28·61-s − 0.610i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.174264735\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.174264735\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 5iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 11iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 5iT - 67T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 7T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−7.946110573982799486091420010926, −7.22707766623033468348974339257, −6.28223934468593390261912801886, −5.78204595803649042373411526002, −4.81062052791549533876991006955, −4.35263356116546961295477839035, −3.09151127709324076511899581190, −2.74170557376387366199668566427, −1.39936001262246305795840570720, −0.32101073849467955773732352010,
1.10947405025221599273854521377, 2.35935998883737133930097634312, 2.79229716152413812994803029694, 4.04248090932884763695847172434, 4.64390732814656640974551887554, 5.36370588995301521812360886280, 6.29586370172444276615365349420, 6.72315927607361845491847168706, 7.59673282857246887469174827491, 8.412412552174874456933094093239