L(s) = 1 | + 2·4-s + (−2.5 + 4.33i)7-s + (4.5 − 2.59i)13-s + 4·16-s + (−3.5 + 6.06i)19-s + (−2.5 + 4.33i)25-s + (−5 + 8.66i)28-s + (2 + 5.19i)31-s + (4.5 + 2.59i)37-s + (1.5 + 0.866i)43-s + (−9.00 − 15.5i)49-s + (9 − 5.19i)52-s − 8.66i·61-s + 8·64-s + (5.5 + 9.52i)67-s + ⋯ |
L(s) = 1 | + 4-s + (−0.944 + 1.63i)7-s + (1.24 − 0.720i)13-s + 16-s + (−0.802 + 1.39i)19-s + (−0.5 + 0.866i)25-s + (−0.944 + 1.63i)28-s + (0.359 + 0.933i)31-s + (0.739 + 0.427i)37-s + (0.228 + 0.132i)43-s + (−1.28 − 2.22i)49-s + (1.24 − 0.720i)52-s − 1.10i·61-s + 64-s + (0.671 + 1.16i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45616 + 0.964131i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45616 + 0.964131i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 + (-2 - 5.19i)T \) |
good | 2 | \( 1 - 2T^{2} \) |
| 5 | \( 1 + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (2.5 - 4.33i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 37 | \( 1 + (-4.5 - 2.59i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.5 - 0.866i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 8.66iT - 61T^{2} \) |
| 67 | \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-12 + 6.92i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (10.5 + 6.06i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−10.38906827642735866738781604118, −9.573793682984379580503469082135, −8.571603791360317168259616792372, −7.966982487468075399162842259388, −6.63472160441853148145316048406, −6.03316770637505305075552413659, −5.48253682978708830472600486482, −3.62708941106990905197283893945, −2.86540413891020270540426299325, −1.71551518566456141501732307859,
0.861651706826499882621914749961, 2.41555362828993481173566442498, 3.65328169563820756027401966491, 4.35630763141934893195528245498, 6.08606564867199397062043696805, 6.60186605864454167863520068670, 7.26491832133683274984688223973, 8.212228305222830744093237394997, 9.366647916484757163315323806657, 10.19946170759739813632001890115