L(s) = 1 | + 1.17i·2-s + 0.611·4-s + (2.16 + 3.75i)5-s + 3.07i·8-s + (−4.42 + 2.55i)10-s + (−1.87 − 1.08i)11-s + (2.25 + 1.30i)13-s − 2.40·16-s + (−0.585 − 1.01i)17-s + (2.09 + 1.20i)19-s + (1.32 + 2.29i)20-s + (1.27 − 2.20i)22-s + (−3.16 + 1.82i)23-s + (−6.88 + 11.9i)25-s + (−1.53 + 2.65i)26-s + ⋯ |
L(s) = 1 | + 0.833i·2-s + 0.305·4-s + (0.968 + 1.67i)5-s + 1.08i·8-s + (−1.39 + 0.807i)10-s + (−0.564 − 0.325i)11-s + (0.624 + 0.360i)13-s − 0.600·16-s + (−0.142 − 0.245i)17-s + (0.480 + 0.277i)19-s + (0.296 + 0.513i)20-s + (0.271 − 0.470i)22-s + (−0.659 + 0.380i)23-s + (−1.37 + 2.38i)25-s + (−0.300 + 0.520i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.218441693\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.218441693\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.17iT - 2T^{2} \) |
| 5 | \( 1 + (-2.16 - 3.75i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.87 + 1.08i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.25 - 1.30i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.585 + 1.01i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.09 - 1.20i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.16 - 1.82i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.589 - 0.340i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6.55iT - 31T^{2} \) |
| 37 | \( 1 + (-2.55 + 4.42i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.68 + 6.38i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.12 + 3.68i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 7.14T + 47T^{2} \) |
| 53 | \( 1 + (2.79 - 1.61i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 5.83T + 59T^{2} \) |
| 61 | \( 1 - 7.17iT - 61T^{2} \) |
| 67 | \( 1 - 6.65T + 67T^{2} \) |
| 71 | \( 1 + 1.95iT - 71T^{2} \) |
| 73 | \( 1 + (-10.3 + 5.95i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 9.75T + 79T^{2} \) |
| 83 | \( 1 + (0.796 + 1.37i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.04 - 5.28i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.36 - 1.36i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.988552742970281738850712418417, −9.187750458761727300088374808446, −8.024986157001982306989187747431, −7.31905685404347796099014542985, −6.71958911987730667433280599261, −5.81726514140804586363153340550, −5.60093929797478544032964382186, −3.82144011550588007574592623496, −2.71326793584704798989664259151, −2.01019925707253228194278190030,
0.910739797646622335747044821396, 1.77468359271810512318162354525, 2.79212955069668639683529801324, 4.10779931041661842932992986012, 5.00058287614088481823599829779, 5.82787928111929386838715284906, 6.66187241281091827346966662480, 7.956616895435633472718968474791, 8.608850697543086146007048261313, 9.579303862472565453394457188342