L(s) = 1 | + (1.40 + 1.01i)2-s + (0.618 − 1.90i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (2.80 − 2.03i)6-s + (−1.07 − 3.29i)7-s + (0.535 − 1.64i)8-s + (−0.809 − 0.587i)9-s − 1.73·10-s + 1.99·12-s + (1.85 − 5.70i)14-s + (0.618 + 1.90i)15-s + (4.04 − 2.93i)16-s + (−5.60 + 4.07i)17-s + (−0.535 − 1.64i)18-s + (2.14 − 6.58i)19-s + ⋯ |
L(s) = 1 | + (0.990 + 0.719i)2-s + (0.356 − 1.09i)3-s + (0.154 + 0.475i)4-s + (−0.361 + 0.262i)5-s + (1.14 − 0.831i)6-s + (−0.404 − 1.24i)7-s + (0.189 − 0.582i)8-s + (−0.269 − 0.195i)9-s − 0.547·10-s + 0.577·12-s + (0.495 − 1.52i)14-s + (0.159 + 0.491i)15-s + (1.01 − 0.734i)16-s + (−1.35 + 0.987i)17-s + (−0.126 − 0.388i)18-s + (0.491 − 1.51i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.11893 - 1.17416i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.11893 - 1.17416i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.40 - 1.01i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.618 + 1.90i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + (1.07 + 3.29i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (5.60 - 4.07i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.14 + 6.58i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (3.23 + 2.35i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.09 - 9.51i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.14 - 6.58i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + (1.85 - 5.70i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-4.85 - 3.52i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.60 + 4.07i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.14 - 6.58i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (5.60 + 4.07i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (14.0 - 10.1i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-8.09 - 5.87i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.68941424567637152040548906874, −9.610438316584992879576383615904, −8.358924519566506946578623575221, −7.35873270120294297058469312420, −6.88636891376923386940730221591, −6.37454550332788648228260722170, −4.84846327501120639091903141712, −4.05509080040572298946124665975, −2.84573552502410712571439142878, −1.01495658695033735725874075568,
2.26971881174473642899523580391, 3.27686441102300989722268628975, 4.04074091337478572509069595982, 5.01219334028428654375452889635, 5.67585136889396041493817410351, 7.19719037527974905094506381152, 8.645676047018287251377581702956, 9.026892371329417206749866559920, 10.00921328508642458454588930020, 11.00727377528648362519635738419