L(s) = 1 | − 1.11·2-s + (0.521 − 0.903i)3-s − 0.754·4-s + (−0.0866 − 0.150i)5-s + (−0.582 + 1.00i)6-s + (−0.0906 + 0.157i)7-s + 3.07·8-s + (0.955 + 1.65i)9-s + (0.0966 + 0.167i)10-s + (0.292 + 0.506i)11-s + (−0.393 + 0.681i)12-s + (0.5 + 0.866i)13-s + (0.101 − 0.175i)14-s − 0.180·15-s − 1.92·16-s + (1.60 − 2.77i)17-s + ⋯ |
L(s) = 1 | − 0.789·2-s + (0.301 − 0.521i)3-s − 0.377·4-s + (−0.0387 − 0.0670i)5-s + (−0.237 + 0.411i)6-s + (−0.0342 + 0.0593i)7-s + 1.08·8-s + (0.318 + 0.551i)9-s + (0.0305 + 0.0529i)10-s + (0.0882 + 0.152i)11-s + (−0.113 + 0.196i)12-s + (0.138 + 0.240i)13-s + (0.0270 − 0.0468i)14-s − 0.0466·15-s − 0.480·16-s + (0.388 − 0.673i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.900609 - 0.268964i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.900609 - 0.268964i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-1.87 - 5.24i)T \) |
good | 2 | \( 1 + 1.11T + 2T^{2} \) |
| 3 | \( 1 + (-0.521 + 0.903i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.0866 + 0.150i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.0906 - 0.157i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.292 - 0.506i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.60 + 2.77i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.80 + 4.86i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 0.834T + 23T^{2} \) |
| 29 | \( 1 - 1.05T + 29T^{2} \) |
| 37 | \( 1 + (-4.25 + 7.36i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.77 + 9.99i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.47 - 6.01i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + (0.437 + 0.757i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.01 - 10.4i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + (1.13 + 1.95i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.93 + 6.81i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.46 - 11.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.979 - 1.69i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.25 + 10.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−10.93137388089503413041082539967, −10.15913104116809836528884807131, −9.174378318644861852567338430045, −8.548078259061787470458386898960, −7.51438348873185465711558070610, −6.95240861302402288362661624054, −5.27564835435016934375728031449, −4.32161684102802177627860678638, −2.58624211014928837990515184211, −1.04146821523292160441572077740,
1.23301997317312583891426779339, 3.35080953429158059630726298592, 4.26599724252389426045701553041, 5.52489969617555961842443774707, 6.85458380543071249859182855501, 8.005344864060351899206094712664, 8.624414437650855410112417984552, 9.754439804539223733015004174111, 9.974153888959312252022423568591, 11.04431659735248804965043277818