L(s) = 1 | + (−1.34 − 2.32i)5-s + (−2.48 + 4.30i)7-s + (1.26 − 2.19i)11-s + (−2.21 − 3.83i)13-s + 2.43·17-s + 4.18·19-s + (−0.570 − 0.988i)23-s + (−1.09 + 1.89i)25-s + (3.00 − 5.20i)29-s + (2.65 + 4.59i)31-s + 13.3·35-s − 0.241·37-s + (3.21 + 5.56i)41-s + (−5.57 + 9.65i)43-s + (−2.37 + 4.11i)47-s + ⋯ |
L(s) = 1 | + (−0.599 − 1.03i)5-s + (−0.939 + 1.62i)7-s + (0.382 − 0.662i)11-s + (−0.614 − 1.06i)13-s + 0.590·17-s + 0.959·19-s + (−0.118 − 0.206i)23-s + (−0.218 + 0.378i)25-s + (0.557 − 0.966i)29-s + (0.476 + 0.825i)31-s + 2.25·35-s − 0.0397·37-s + (0.501 + 0.868i)41-s + (−0.849 + 1.47i)43-s + (−0.346 + 0.600i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2077333300\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2077333300\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.34 + 2.32i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2.48 - 4.30i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.26 + 2.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.21 + 3.83i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 2.43T + 17T^{2} \) |
| 19 | \( 1 - 4.18T + 19T^{2} \) |
| 23 | \( 1 + (0.570 + 0.988i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.00 + 5.20i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.65 - 4.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.241T + 37T^{2} \) |
| 41 | \( 1 + (-3.21 - 5.56i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.57 - 9.65i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.37 - 4.11i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 9.38T + 53T^{2} \) |
| 59 | \( 1 + (5.40 + 9.36i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.16 - 7.20i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.13 + 1.96i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.52T + 71T^{2} \) |
| 73 | \( 1 + 3.34T + 73T^{2} \) |
| 79 | \( 1 + (-3.14 + 5.44i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.738 + 1.27i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + (-5.89 + 10.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.183384614515432981256693018321, −7.84251175120834862760588022373, −6.50475880216188513291032099898, −5.91260662849047430699545449677, −5.21122831389031435772415964982, −4.51752808124185457393620331973, −3.10840024567238783865153136557, −2.93417392143305974817684032438, −1.27798781256043520886734150105, −0.06842960997739887896908221119,
1.36593190528152791893678747467, 2.78031156465043554137694943268, 3.60656381972858555720677169569, 4.07602902449550146906474717682, 5.02986039311663461150255594065, 6.33213374841964900439702540029, 6.98713410360720842889522646467, 7.23194533721261184158506967334, 7.87987778645012914585079967820, 9.206358317360224344685970024722