L(s) = 1 | + i·2-s − 4-s + (−2 + i)5-s + 0.828i·7-s − i·8-s + (−1 − 2i)10-s + 0.828·11-s − 4.82i·13-s − 0.828·14-s + 16-s − 0.828i·17-s + (2 − i)20-s + 0.828i·22-s + 8.48i·23-s + (3 − 4i)25-s + 4.82·26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.894 + 0.447i)5-s + 0.313i·7-s − 0.353i·8-s + (−0.316 − 0.632i)10-s + 0.249·11-s − 1.33i·13-s − 0.221·14-s + 0.250·16-s − 0.200i·17-s + (0.447 − 0.223i)20-s + 0.176i·22-s + 1.76i·23-s + (0.600 − 0.800i)25-s + 0.946·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.196530882\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.196530882\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 - 0.828iT - 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 + 4.82iT - 13T^{2} \) |
| 17 | \( 1 + 0.828iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8.48iT - 23T^{2} \) |
| 29 | \( 1 - 9.65T + 29T^{2} \) |
| 37 | \( 1 + 10.4iT - 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 - 9.65iT - 43T^{2} \) |
| 47 | \( 1 - 5.65iT - 47T^{2} \) |
| 53 | \( 1 - 0.343iT - 53T^{2} \) |
| 59 | \( 1 - 3.17T + 59T^{2} \) |
| 61 | \( 1 - 0.828T + 61T^{2} \) |
| 67 | \( 1 - 9.17iT - 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 - 13.6iT - 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 1.65iT - 83T^{2} \) |
| 89 | \( 1 - 4.82T + 89T^{2} \) |
| 97 | \( 1 + 11.3iT - 97T^{2} \) |
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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
−8.779388469647855640443026295036, −8.168176443053394214281063968078, −7.53011411721337552580795442365, −6.92948303927166326160429503970, −5.98340164432540046319805181445, −5.32336525289576685220608852465, −4.38254680231810686805080187457, −3.48353639321229734455327677952, −2.73213947864518013787378141391, −0.952230693505777515026116116314,
0.51147160771412702295183990121, 1.66357846160614310160657713703, 2.83884163364883867871465010789, 3.85565269632063985398092528753, 4.46290480656056156750578325378, 5.05201035346064229935749506238, 6.51569878469765960582817102871, 6.91680649136159838301424315915, 8.142961286426858112579303710644, 8.567011612158976203459154134747