L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·6-s + (−0.330 − 0.573i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s − 4.57·11-s + (−0.499 + 0.866i)12-s + (0.0966 + 0.167i)13-s + 0.661·14-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−3.61 + 6.26i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + 0.408·6-s + (−0.125 − 0.216i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + 0.316·10-s − 1.38·11-s + (−0.144 + 0.249i)12-s + (0.0268 + 0.0464i)13-s + 0.176·14-s + (−0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.877 + 1.51i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8580489018\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8580489018\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (1.42 + 5.91i)T \) |
good | 7 | \( 1 + (0.330 + 0.573i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 4.57T + 11T^{2} \) |
| 13 | \( 1 + (-0.0966 - 0.167i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.61 - 6.26i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.32 - 4.02i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 - 7.49T + 29T^{2} \) |
| 31 | \( 1 - 5.82T + 31T^{2} \) |
| 41 | \( 1 + (3.52 + 6.10i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 9.15T + 43T^{2} \) |
| 47 | \( 1 + 2.70T + 47T^{2} \) |
| 53 | \( 1 + (2.71 - 4.70i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.68 + 6.38i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.29 - 10.9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.20 - 9.00i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.24 - 7.35i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 3.32T + 73T^{2} \) |
| 79 | \( 1 + (0.0771 + 0.133i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.82 - 15.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.44 + 2.50i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5.31T + 97T^{2} \) |
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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
−10.10782590383608124888577846114, −8.769855133273111911885833686702, −8.311112437715213364965897170019, −7.49333501274264250904690656314, −6.72524288913348797719674160132, −5.78390282385950484604115562819, −5.06004848324582134034275367538, −3.97138020593397619269927571216, −2.44325174269202394435130641515, −0.971601163879615362915155230171,
0.57668769483507355944681932856, 2.70614686807533374372424312208, 3.00851107505951735999924962716, 4.70459451737824533737483446560, 5.01165010764073044802572181785, 6.48110990140603906671847599040, 7.29077029501105647100431869240, 8.246913171617134132822905113214, 9.097798951282687988406007462927, 9.829202882625575088385467684383