L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.09 + 1.62i)7-s − 0.999·8-s + (−2.59 + 1.5i)11-s − 2.44·13-s + (2.44 − 0.999i)14-s + (−0.5 + 0.866i)16-s + (−0.878 + 0.507i)17-s + (0.878 + 0.507i)19-s + 3i·22-s + (2.12 − 3.67i)23-s + (−1.22 + 2.12i)26-s + (0.358 − 2.62i)28-s − 1.24i·29-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.790 + 0.612i)7-s − 0.353·8-s + (−0.783 + 0.452i)11-s − 0.679·13-s + (0.654 − 0.267i)14-s + (−0.125 + 0.216i)16-s + (−0.213 + 0.123i)17-s + (0.201 + 0.116i)19-s + 0.639i·22-s + (0.442 − 0.766i)23-s + (−0.240 + 0.416i)26-s + (0.0677 − 0.495i)28-s − 0.230i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.021456402\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.021456402\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.09 - 1.62i)T \) |
good | 11 | \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 + (0.878 - 0.507i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.878 - 0.507i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.12 + 3.67i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.24iT - 29T^{2} \) |
| 31 | \( 1 + (-4.86 + 2.80i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.13 - 4.12i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.02T + 41T^{2} \) |
| 43 | \( 1 - 8.24iT - 43T^{2} \) |
| 47 | \( 1 + (-0.878 - 0.507i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.621 - 1.07i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.76 - 9.98i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.12 - 2.95i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.66 + 5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.2iT - 71T^{2} \) |
| 73 | \( 1 + (-4.18 - 7.24i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.62 - 9.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.16iT - 83T^{2} \) |
| 89 | \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.76T + 97T^{2} \) |
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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
−8.636287439451338958018552153677, −8.081143975961691164076867659338, −7.27107083840131855312058733498, −6.26310410642324739631359283171, −5.43648378769008941265248057609, −4.75661668803651793189988258412, −4.16977149879214274631260438910, −2.69066527785227089840014339626, −2.40728125170721828775730007720, −1.07181392900975254838991873451,
0.63684614533676459896866628468, 2.14345395145789943621590475316, 3.19233590744202983588730701914, 4.13302382709088214320641945190, 5.00924228799953951710097394554, 5.40806200411369790685704353832, 6.48044358924271194192160944862, 7.27592677301553890381451125440, 7.77754495055403280369455936370, 8.441939236013342183232629020959