| L(s) = 1 | + (−0.405 − 2.55i)2-s + (2.54 + 1.29i)3-s + (−4.47 + 1.45i)4-s + (1.77 − 1.35i)5-s + (2.28 − 7.03i)6-s + (0.595 − 0.595i)7-s + (3.18 + 6.25i)8-s + (3.03 + 4.17i)9-s + (−4.18 − 4.00i)10-s + (1.90 + 1.38i)11-s + (−13.2 − 2.10i)12-s + (0.574 − 3.62i)13-s + (−1.76 − 1.28i)14-s + (6.28 − 1.13i)15-s + (7.08 − 5.15i)16-s + (1.19 + 2.33i)17-s + ⋯ |
| L(s) = 1 | + (−0.286 − 1.80i)2-s + (1.46 + 0.748i)3-s + (−2.23 + 0.727i)4-s + (0.795 − 0.605i)5-s + (0.933 − 2.87i)6-s + (0.225 − 0.225i)7-s + (1.12 + 2.21i)8-s + (1.01 + 1.39i)9-s + (−1.32 − 1.26i)10-s + (0.575 + 0.418i)11-s + (−3.83 − 0.607i)12-s + (0.159 − 1.00i)13-s + (−0.471 − 0.342i)14-s + (1.62 − 0.293i)15-s + (1.77 − 1.28i)16-s + (0.288 + 0.566i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.222 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25477 - 1.57317i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25477 - 1.57317i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.77 + 1.35i)T \) |
| 19 | \( 1 + (3.03 - 3.12i)T \) |
| good | 2 | \( 1 + (0.405 + 2.55i)T + (-1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (-2.54 - 1.29i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-0.595 + 0.595i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.90 - 1.38i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.574 + 3.62i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.19 - 2.33i)T + (-9.99 + 13.7i)T^{2} \) |
| 23 | \( 1 + (-0.680 - 4.29i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (2.86 + 8.80i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (9.37 + 3.04i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.24 - 0.672i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (1.33 + 1.83i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (2.82 + 2.82i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.79 + 0.913i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (6.32 + 3.22i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (4.32 - 3.13i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-9.09 - 6.60i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.21 + 2.14i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (7.01 - 2.27i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.71 - 10.8i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-5.20 - 16.0i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (12.2 - 6.25i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-0.266 - 0.193i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.53 - 4.98i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−10.44284513059520054248597988520, −9.800810573801923212281757312114, −9.380348529750234654626816025364, −8.485547180589461656279295148191, −7.86909889399659704885279875688, −5.55426589057321587713416974849, −4.22593124929278887309711726003, −3.69151313739272501096704672697, −2.43836522063637074755352997256, −1.54801432944169176660606584866,
1.79174663958212942285977817025, 3.35022149307955290767693647923, 4.86857065747162974412300659743, 6.19472275251094228302284317772, 6.87453432017045215589254466569, 7.41599527200466718934260546788, 8.619474894257081388809298526124, 8.968872437557976536196489218173, 9.629982843797437057304296101457, 11.06853403193723408599154895477
