L(s) = 1 | + (0.840 + 1.13i)2-s + (−0.588 + 1.91i)4-s − 2.67i·5-s + (0.370 − 2.61i)7-s + (−2.66 + 0.935i)8-s + (3.04 − 2.25i)10-s + 1.39i·11-s − 3.08i·13-s + (3.29 − 1.77i)14-s + (−3.30 − 2.25i)16-s − 5.83i·17-s + 1.91·19-s + (5.12 + 1.57i)20-s + (−1.58 + 1.17i)22-s − 1.39i·23-s + ⋯ |
L(s) = 1 | + (0.593 + 0.804i)2-s + (−0.294 + 0.955i)4-s − 1.19i·5-s + (0.140 − 0.990i)7-s + (−0.943 + 0.330i)8-s + (0.963 − 0.711i)10-s + 0.421i·11-s − 0.855i·13-s + (0.879 − 0.475i)14-s + (−0.826 − 0.562i)16-s − 1.41i·17-s + 0.440·19-s + (1.14 + 0.352i)20-s + (−0.338 + 0.250i)22-s − 0.291i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.78223 - 0.397885i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78223 - 0.397885i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.840 - 1.13i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.370 + 2.61i)T \) |
good | 5 | \( 1 + 2.67iT - 5T^{2} \) |
| 11 | \( 1 - 1.39iT - 11T^{2} \) |
| 13 | \( 1 + 3.08iT - 13T^{2} \) |
| 17 | \( 1 + 5.83iT - 17T^{2} \) |
| 19 | \( 1 - 1.91T + 19T^{2} \) |
| 23 | \( 1 + 1.39iT - 23T^{2} \) |
| 29 | \( 1 + 4.90T + 29T^{2} \) |
| 31 | \( 1 - 9.53T + 31T^{2} \) |
| 37 | \( 1 - 6.83T + 37T^{2} \) |
| 41 | \( 1 + 0.693iT - 41T^{2} \) |
| 43 | \( 1 + 0.678iT - 43T^{2} \) |
| 47 | \( 1 + 8.26T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 12.8iT - 61T^{2} \) |
| 67 | \( 1 - 3.08iT - 67T^{2} \) |
| 71 | \( 1 - 9.79iT - 71T^{2} \) |
| 73 | \( 1 + 5.91iT - 73T^{2} \) |
| 79 | \( 1 - 12.0iT - 79T^{2} \) |
| 83 | \( 1 - 5.77T + 83T^{2} \) |
| 89 | \( 1 - 6.65iT - 89T^{2} \) |
| 97 | \( 1 + 12.8iT - 97T^{2} \) |
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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
−10.07120359428524304586046591044, −9.297781187507714618568576358427, −8.324387428471227741208371099971, −7.64061463134765161764152552757, −6.88363955873719650011375484120, −5.65189012904296219317201595676, −4.83853527809118935526218970916, −4.23507060983871723528366422677, −2.90606093386302356053304316954, −0.798657005389844421301671526634,
1.79587616647147353040132944317, 2.81570609734148008068316004894, 3.69364707887566834923848234558, 4.88118729958732558392088128219, 6.10469857172972575280318175810, 6.42575137793156901219640688122, 7.87936532079055829732675298138, 8.941055115008793670953830697943, 9.754745283201737737758424418072, 10.60473777458291136511135184875