| L(s) = 1 | + (−1.81 + 0.846i)2-s + (0.00445 + 1.73i)3-s + (1.29 − 1.54i)4-s + (−1.47 − 3.14i)6-s + (2.21 + 0.193i)7-s + (−0.00834 + 0.0311i)8-s + (−2.99 + 0.0154i)9-s + (1.09 − 0.193i)11-s + (2.68 + 2.23i)12-s + (1.28 − 2.74i)13-s + (−4.18 + 1.52i)14-s + (0.688 + 3.90i)16-s + (1.56 + 5.82i)17-s + (5.43 − 2.56i)18-s + (3.84 − 2.22i)19-s + ⋯ |
| L(s) = 1 | + (−1.28 + 0.598i)2-s + (0.00257 + 0.999i)3-s + (0.647 − 0.772i)4-s + (−0.602 − 1.28i)6-s + (0.836 + 0.0731i)7-s + (−0.00295 + 0.0110i)8-s + (−0.999 + 0.00514i)9-s + (0.330 − 0.0582i)11-s + (0.773 + 0.645i)12-s + (0.355 − 0.762i)13-s + (−1.11 + 0.406i)14-s + (0.172 + 0.976i)16-s + (0.378 + 1.41i)17-s + (1.28 − 0.605i)18-s + (0.882 − 0.509i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.402215 + 0.726481i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.402215 + 0.726481i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.00445 - 1.73i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (1.81 - 0.846i)T + (1.28 - 1.53i)T^{2} \) |
| 7 | \( 1 + (-2.21 - 0.193i)T + (6.89 + 1.21i)T^{2} \) |
| 11 | \( 1 + (-1.09 + 0.193i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.28 + 2.74i)T + (-8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (-1.56 - 5.82i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.84 + 2.22i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.293 - 3.34i)T + (-22.6 + 3.99i)T^{2} \) |
| 29 | \( 1 + (1.74 + 0.635i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-7.13 - 5.98i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-4.53 + 1.21i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (1.07 + 2.95i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-5.86 - 4.10i)T + (14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (1.10 - 12.5i)T + (-46.2 - 8.16i)T^{2} \) |
| 53 | \( 1 + (6.34 + 6.34i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.242 + 1.37i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.30 - 1.93i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (8.07 + 3.76i)T + (43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (6.78 + 3.91i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.167 - 0.0447i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.697 - 1.91i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.27 - 4.86i)T + (-53.3 + 63.5i)T^{2} \) |
| 89 | \( 1 + (-6.70 - 11.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.92 - 14.1i)T + (-33.1 - 91.1i)T^{2} \) |
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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
−10.63971104532910040926890830592, −9.708961503602765643992315463119, −9.107504546062321893059577263680, −8.145548473565702462118406843037, −7.83193557802939905889294336177, −6.40071485553476842431256439217, −5.54404941821187812914367414411, −4.39680454445001114799458923953, −3.23517503491066428315650642973, −1.26778939072783134971708857925,
0.825743567872831859961135446611, 1.82294696441029028923902961750, 2.91691676259609199071276189177, 4.69496354833295137800920614892, 5.90194113605186895759524105084, 7.13230174399254469281115996869, 7.73957433110020436956054093381, 8.523652506328244276916473904989, 9.268504977457546362966306937748, 10.10023023343497473190192801722
