| L(s) = 1 | + (1.41 − 0.0477i)2-s + (−0.942 − 0.135i)3-s + (1.99 − 0.135i)4-s + (0.224 + 0.765i)5-s + (−1.33 − 0.146i)6-s + (−0.326 − 0.376i)7-s + (2.81 − 0.286i)8-s + (−2.00 − 0.589i)9-s + (0.354 + 1.07i)10-s + (−1.90 − 1.22i)11-s + (−1.89 − 0.143i)12-s + (−1.73 + 2.00i)13-s + (−0.479 − 0.516i)14-s + (−0.108 − 0.751i)15-s + (3.96 − 0.539i)16-s + (−1.98 + 0.905i)17-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.0337i)2-s + (−0.544 − 0.0782i)3-s + (0.997 − 0.0675i)4-s + (0.100 + 0.342i)5-s + (−0.546 − 0.0597i)6-s + (−0.123 − 0.142i)7-s + (0.994 − 0.101i)8-s + (−0.669 − 0.196i)9-s + (0.112 + 0.338i)10-s + (−0.574 − 0.369i)11-s + (−0.548 − 0.0412i)12-s + (−0.482 + 0.556i)13-s + (−0.128 − 0.138i)14-s + (−0.0279 − 0.194i)15-s + (0.990 − 0.134i)16-s + (−0.480 + 0.219i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40027 - 0.0338170i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40027 - 0.0338170i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 + 0.0477i)T \) |
| 23 | \( 1 + (2.43 + 4.13i)T \) |
| good | 3 | \( 1 + (0.942 + 0.135i)T + (2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (-0.224 - 0.765i)T + (-4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (0.326 + 0.376i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (1.90 + 1.22i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (1.73 - 2.00i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.98 - 0.905i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (1.56 - 3.43i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.412 - 0.902i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-8.24 + 1.18i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.88 + 6.41i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (0.272 - 0.0801i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.81 - 12.6i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 0.0264iT - 47T^{2} \) |
| 53 | \( 1 + (5.21 - 4.51i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.01 - 3.47i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-8.30 + 1.19i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (1.06 - 0.686i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (5.66 + 8.82i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (4.66 - 10.2i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-5.81 + 6.71i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (17.3 + 5.08i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (10.4 + 1.49i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-3.67 - 12.5i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−14.15616949794337690993987112059, −12.95301036984194121899663963058, −12.02293898529706273523485926172, −11.08269696490849009600886569744, −10.17416158630294416852584354158, −8.266860655156001578992025159818, −6.71986876664506850457695254056, −5.89623434296339140788980576892, −4.47589735441809527750688801498, −2.72022061069256294946255008388,
2.73238896738063104631392610839, 4.72931093472403117142741203320, 5.58373407417338649484738983898, 6.90072600899731393482580093593, 8.323227916322011163131249332801, 10.06265001452783619027655582673, 11.16110846068703049679754279217, 12.05807906778237534151382487446, 13.03447997923740237272128829719, 13.90364134826985496703006693205
