L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.809 + 0.587i)3-s + (0.809 − 0.587i)4-s + (−0.618 + 1.90i)5-s + (0.309 − 0.951i)6-s + (3.23 − 2.35i)7-s + (−2.42 − 1.76i)8-s + (0.309 + 0.951i)9-s + 1.99·10-s + 12-s + (0.618 + 1.90i)13-s + (−3.23 − 2.35i)14-s + (−1.61 + 1.17i)15-s + (−0.309 + 0.951i)16-s + (0.618 − 1.90i)17-s + (0.809 − 0.587i)18-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.467 + 0.339i)3-s + (0.404 − 0.293i)4-s + (−0.276 + 0.850i)5-s + (0.126 − 0.388i)6-s + (1.22 − 0.888i)7-s + (−0.858 − 0.623i)8-s + (0.103 + 0.317i)9-s + 0.632·10-s + 0.288·12-s + (0.171 + 0.527i)13-s + (−0.864 − 0.628i)14-s + (−0.417 + 0.303i)15-s + (−0.0772 + 0.237i)16-s + (0.149 − 0.461i)17-s + (0.190 − 0.138i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.53473 - 0.619573i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53473 - 0.619573i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.309 + 0.951i)T + (-1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (0.618 - 1.90i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-3.23 + 2.35i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-0.618 - 1.90i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.618 + 1.90i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + (4.85 - 3.52i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.47 + 7.60i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (4.85 - 3.52i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (1.61 + 1.17i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + (6.47 + 4.70i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.85 - 5.70i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.23 + 2.35i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.85 - 5.70i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (11.3 - 8.22i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.23 - 3.80i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (3.70 - 11.4i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-0.618 - 1.90i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.10211445040991423154753015562, −10.68630404786237188350348878147, −9.695362117651877324787003615298, −8.708445421855586774809953100062, −7.40874187366503980935469863211, −6.89623714302129936880438135400, −5.25280233912964627661951953305, −3.94845719117174148511132884871, −2.87592822451911397101739816204, −1.49633516232260861168588190176,
1.71684100947297259348320137401, 3.15610091110618463559268987797, 4.87407931774073950344897226345, 5.72847286849391974811638868002, 7.04087804897277493968612192947, 7.959097778485646671065223030361, 8.567982290302422934725276694513, 9.077796458767855077568000040159, 10.81649705952253333475672343055, 11.67805846077870929938728622217