L(s) = 1 | + 2i·7-s − 6·11-s − 5i·13-s − 3i·17-s − 2·19-s − 6i·23-s − 3·29-s − 4·31-s + 5i·37-s − 6·41-s + 10i·43-s + 3·49-s + 6i·53-s + 12·59-s + 5·61-s + ⋯ |
L(s) = 1 | + 0.755i·7-s − 1.80·11-s − 1.38i·13-s − 0.727i·17-s − 0.458·19-s − 1.25i·23-s − 0.557·29-s − 0.718·31-s + 0.821i·37-s − 0.937·41-s + 1.52i·43-s + 0.428·49-s + 0.824i·53-s + 1.56·59-s + 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9942595267\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9942595267\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 5iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.043646670752233854528589348122, −7.39595517181118366384425852162, −6.52249938862100524283843083494, −5.69256116737658238253300917196, −5.23292382961246028370071152511, −4.66208884705078039334772759680, −3.43986437011666863051361522215, −2.69085462913392214965185512487, −2.25546324494438266553196219455, −0.70567478024749120046517004194,
0.31566750087458468002636643968, 1.77257783546612488084048496557, 2.31876788625713483086797959618, 3.63130951905797775119081344365, 3.94209099472952464764194416203, 5.04607003206315903452143038662, 5.45958955349260255145868380700, 6.39289644502238233384097934576, 7.21591532347231014218535267455, 7.51286835482754762533081633188